Q&A: statistics

The average score is useful, but not sufficient on its own. The reason is that a rating from 1 to 5 is an ordinal scale: you can say 5 is better than 4 and 4 is better than 3. But there is no guarantee that the "distance" between 4 and 5 equals the distance between 1 and 2. And the mean relies precisely on this "equality of distances".

Moreover, the mean hides the distribution. For the two branches below the same mean of 3.0 comes out:

Both have a mean of 3.0, but reality is entirely different: in branch "A" everyone says "average", in "B" half are very satisfied and half very dissatisfied. In "B" the median also comes out as 3 (the 10th value is 1, the 11th value is 5, the middle is 3), even though no customer actually gave a 3.

Your data (compiled by the author, 20 customers):

Calculation (checked with Python): mean = (1×2 + 2×3 + 3×3 + 4×6 + 5×6) ÷ 20 = 71 ÷ 20 = 3.55. Median = 4 (the 10th and 11th values are both 4). There are two modes: 4 and 5 (6 times each). Customers who gave 4–5 points: 12, that is, 60%. Those who gave 1–2 points: 5, that is, 25%.

If a report is written like this, it gives the manager more benefit: "60% of customers gave 4 or 5 points, 25% gave 1 or 2 points, median 4, mean 3.55". This is more precise than a single "3.55".

Rules: 1. Always show the distribution of scores (what percentage gave each score). 2. The median, as well as the share of the top two scores (4–5) and the share of the bottom two scores (1–2), are indicators suited to an ordinal scale. 3. Give the mean as a supplementary indicator, especially when comparing over time (last month 3.40, this month 3.55). But do not make it the sole basis when the number of customers is small. 4. Show customers who did not answer or said "cannot say" separately: do not add them to the mean as 0.

Source: general statistical rules (measurement scales, mean, median, mode). The data were compiled by the author.

Such a conclusion resembles "survivorship bias". You are only looking at the cafés that did not close. Among those that closed there may also have been many cafés with large halls and expensive interiors, but they are no longer visible. To find the reason for success, the survivors must be compared with those that closed.

Example (prepared by the author, hypothetical figures). Three years ago 50 cafés opened in the city. Today 20 of them are operating and 30 have closed.

Among the survivors, those with large halls are 70% (14 out of 20) — this is your observation. But among those that closed they are also exactly 70% (21 out of 30). That is, because there were many cafés with large halls, there are many of them among the survivors too. The probability of survival is the same in both groups — 40%. According to these data, the size of the hall does not predict success.

This mistake is common in business:

1. Reading only the biographies of successful entrepreneurs and saying "they get up early, so getting up early makes you rich".
2. Surveying only the remaining customers and concluding "everyone is satisfied": the dissatisfied have already left.
3. Analysing only the goods currently on sale and saying "goods in this category sell well": the ones that did not sell well have already been removed from the range.

How to protect yourself:

1. Always ask: "Who did not make it onto this list?"
2. Try to find information about those that closed, left or were removed too: old lists, archives, conversations with former customers.
3. Always calculate the share relative to the starting group: not "how many among the survivors" but "how many of those who started survived".

So far there are no grounds to say so. The statistical phenomenon of "regression to the mean" may have been at work here. Each month's result consists of two parts: the branch's permanent level and random factors, that is, circumstances specific to that month that need not repeat the next month. A branch that shows a very bad result in one month has often simply been unlucky. The next month chance may not be so bad, and the result moves back towards its usual level without any measures at all. With the best result it is the other way round.

Example (prepared by the author, hypothetical figures). The average annual plan fulfilment of six branches is around 100%. Results for two months:

The average of the "worst three" rose from 81.7% to 94.3%, and that of the "best three" fell from 115.7% to 104%. But the overall average of the six branches barely changed: 98.7% and 99.2%. Every branch simply moved closer to its usual level.

How to check whether the measures really worked:

1. Compare the result not with a single month but with the branch's own long-term average.
2. Set up a comparison group: apply the measure to some branches with similar results and not to others. If both groups improve equally, the cause is not the measure.
3. Observe for several months: if the measure really has an effect, the improvement will be lasting.

This phenomenon occurs elsewhere too: the day after the day with the most rejects is better, and a new advertising campaign looks "effective" because it started after the worst month. Before making a decision, ask: "Would it not have happened anyway, even if we had done nothing?"

This question can be answered with a control chart. The idea is simple: even when a process is stable, the daily reject share fluctuates randomly. Statistics calculates the usual limits of this fluctuation. If a result goes outside the limits or shows a strange pattern, it is suspected that something in the process has changed.

A simple formula for the reject share:

• average share p = 0.03;
• daily sample n = 400;
• standard deviation = √(p × (1 − p) ÷ n) = √(0.03 × 0.97 ÷ 400) ≈ 0.0085, that is, 0.85 percentage points.

In a widely used approach, the control limits are set at a distance of ± 3 standard deviations from the average.

Example (prepared by the author, hypothetical figures). For the "Temir usta" workshop:

Today's 20 rejects are below the upper control limit (22), but above the warning level (19). So there are not enough grounds to stop the machine at once, but attention should be stepped up: watch tomorrow's result and check the work shift, the raw material batch and the condition of the tools.

Another useful sign is the sequence of results. If the process has not changed, the chances of being above or below the average on any day are roughly equal. The probability of being above the average 8 days in a row is 0.5⁸ ≈ 0.4%. If that happens, the process may have deteriorated even if no single result goes outside the limits.

Going below the lower limit is also a signal: if there are very few rejects, it may mean the inspector has become inattentive or a new, better method is working. That should be checked too.

Practical tip: in Excel, plot the daily reject share and the three lines (average, upper and lower limit) on one chart and hang it on the workshop wall. The foremen will see for themselves when to worry.

Usually this is not an error but the result of rounding. Each share is rounded separately, and a small difference appears in each. When they are added up, the total can differ from 100% by one or two percentage points.

Example (prepared by the author, hypothetical figures). The monthly revenue of the "Lola savdo" shop is 250 million soums:

Three equal parts, on the other hand, are 33.33...% each and are rounded to 33%, giving a total of 99%.

The same thing happens with amounts. If three expenses are 1.4 million soums each, the total is 4.2 million. If "million soums" in the table is rounded to whole numbers: 1 + 1 + 1, while the total is 4. The rows add up to 3, but the "Total" row shows 4.

How to deal with this:

1. Write a note under the table: "Totals may differ from 100% due to rounding".
2. Show one decimal place: 40.6% + 30.6% + 28.8% = 100.0%. Often this alone is enough.
3. If the total in a chart must be exactly 100%, use the largest remainder method: round everything down (40 + 30 + 28 = 98), then add the missing 2 percentage points, one each, to the two with the largest fractional parts: drinks (0.84) and food (0.60); the fractional part of household chemicals (0.56) is the smallest. Result: 41%, 30%, 29% — a total of 100%.
4. Never calculate the "Total" by adding the rounded rows: calculate it from the exact amount and then round it.

This is especially important in accounting reports: if a report is prepared in thousands of soums, it is advisable to set out in advance in the accounting policy a method of adjusting the difference in one line item, so that the assets and liabilities of the balance sheet do not end up differing by a few units because of rounding.

One sentence is enough to explain it to the manager: "Each figure has been rounded separately; the exact figures add up to 100%."

No, it does not. This is one of the most widespread confusions in statistics. "It finds 96 percent of real errors" and "96 percent of flagged receipts are real errors" are completely different things. The second depends on how common erroneous receipts are in general and on how often the software mistakenly flags clean receipts.

Example (prepared by the author, hypothetical figures). "Barakali do'kon" has 10,000 receipts a month. About 50 of them are real errors (0.5%). The software finds 96 percent of real errors, but also mistakenly flags 2 percent of clean receipts as suspicious.

Of the 247 flagged receipts, only 48 are real errors: 48 ÷ 247 ≈ 19.4%. That is, roughly one in five receipts flagged by the software is an error, not 96%.

Why is this so? There are very many clean receipts (9,950), so even a small 2 percent of them gives 199 false signals. That is four times more than the real errors.

Practical conclusions:

1. Do not treat a flagged receipt as an error straight away and blame the salesperson. It is merely in the queue for a first check.
2. When evaluating software, ask for two indicators: what share of real errors it finds and what share of clean receipts it flags by mistake.
3. Estimate in advance how much work the inspecting employee will get: 247 receipts a month have to be checked, 199 of them in vain.
4. If the share of erroneous receipts rises (for example, when a new cashier arrives), the share of real errors among flagged receipts rises too. That is, the same software looks "accurate" to different degrees under different conditions.

This logic applies to any automated check: how many real errors there are among flagged transactions depends on how rare errors are in general.

This is called sales funnel analysis. At each stage some people "drop out". The conversion of each stage is the share of those who move on to the next stage:

Conversion = number of people at the next stage : number of people at the previous stage

The overall result equals the product of the conversions of all stages. So an improvement at one stage affects the final result through the whole chain.

Example (prepared by the author, hypothetical figures). On the "Ipak yo'li kitoblari" website, the average purchase is 250,000 soums.

Now let us calculate what happens if we improve each stage separately by 5 percentage points:

The biggest effect is at the stage of moving from the cart to payment. The reason: the lower the conversion, the larger the relative growth the same 5 percentage points give (from 25% to 30% is one fifth more). Check what is stopping customers at this stage: does the delivery cost pop up only at the end, is registration compulsory, is the form long?

Practical tips:

1. Measure the funnel every month with the same definitions, otherwise the months cannot be compared.
2. Split the stages by customer type as well: the conversion of visitors on phones and on computers can differ greatly.
3. After making a change, observe for long enough to separate the result from random fluctuations.

This kind of test is called an A/B test. The difference between 5% and 6.5% looks large, but with a few dozen clicks chance can produce such a difference too. To check this, statistics uses a test for the difference between two proportions.

The idea: first we assume that "there is actually no difference" (the null hypothesis) and measure how unexpected the observed difference is under that assumption.

Example (prepared by the author, hypothetical figures). The ad test of the "Chimyon sayohat" firm:

Usually, to call a difference "not chance" at a 95% confidence level, the absolute value of z must be greater than 1.96. 1.44 does not reach this threshold. So for now we cannot say with certainty that "B is better". Nor does this mean "B is worse": there is not enough data.

How many people were needed? To reliably detect a difference of 1.5 percentage points (95% confidence, 80% probability of detecting the difference), each group needs approximately:

(1.96 + 0.84)² × (0.05 × 0.95 + 0.065 × 0.935) : 0.015² ≈ 3,773 people.

If each text is seen by 4,000 people and the proportions stay the same, z ≈ 2.88 and the difference is considered reliable.

Rules for a proper test:

1. Show both texts at the same time to a randomly selected audience, otherwise differences between days or weeks will distort the result.
2. Calculate the number of people needed before the test and do not stop it on the day the result "looks good".
3. Besides clicks, track the next step too: a text that gets many clicks may lead to few purchases.

If the data is grouped into intervals, the values within each interval are assumed to lie at its midpoint. After that an ordinary weighted average is calculated:

Average ≈ Σ (interval midpoint × number of people) : total people

This is not an exact value but a good estimate. The narrower the intervals, the more accurate the estimate.

Example (prepared by the author, hypothetical figures). 125 members of the "Baquvvat" gym. The last group was written open-ended in the questionnaire as "over 60"; we took it as equal in width to the previous group, that is, 60–75 years.

Average ≈ 3,935 : 125 ≈ 31.5 years.

The median (the age below which half of the members are) can also be found from a grouped table. Half of 125 is 62.5. The first group has 34 people, so the median is in the second group (25–35). There we need to count another 62.5 − 34 = 28.5 people:

Median ≈ 25 + 28.5 : 58 × 10 ≈ 29.9 years.

The average (31.5) is greater than the median (29.9) because a few members who are much older pull the average up. When planning advertising and the class timetable, the median better represents the "typical" member. Small groups should be looked at separately: in this example there are 12 people over 45 (9.6%), and you might consider whether they need a separate class time.

Tip: if possible, ask for the exact age (or year of birth) in the questionnaire. You can group the data any way you like later, but exact values cannot be recovered from grouped data.

Not so fast. The "average of past days × days in the month" method assumes that every day sells the same. If your sales on weekends are different, the forecast changes noticeably depending on how many such days fell in the past 12 days.

A more accurate method is to give the days a "weight". From the data of your own past months, you find how many times more a weekend day sells compared with an ordinary day, and count the days with that weight.

Example (prepared by the author, hypothetical figures). At the "Yunusobod pishiriqlari" café, data from past months show that Saturday and Sunday bring in 1.5 times the revenue of an ordinary day. The month has 30 days: 22 working days and 8 weekend days. The past 12 days included 8 working days and 4 weekend days.

The simple calculation says 115 million, the weighted one 111.7 million. The reason: half of the 8 weekend days have already passed in the first 12 days of the month, and only 4 remain in the other 18 days. So until the end of the month each day will on average bring in less revenue than before.

What to watch out for:

1. Update the forecast every week: the closer you get to the end of the month, the more accurate it becomes.
2. Take into account not only days of the week but also events specific to the month: mark every day in past months when revenue differed sharply from usual and write down the reason.
3. If the forecast says "not enough", it is a signal to act: in the remaining 18 days you need about 1.29 million : 18 ≈ 71 thousand soums of extra revenue a day, which can be planned through a promotion or a new dish.

Not quite. Finding no faulty lamps in 20 checks does not show that the batch has no faulty ones, only that their share is not very large. This can be checked with probability.

If the share of faulty lamps in the batch is q, the probability that one randomly chosen lamp works is (1 − q). The probability that all 20 work is approximately (1 − q)²⁰. Because the batch is large (1,000 units), this approximation is accurate enough.

Example (prepared by the author, hypothetical figures). The calculation for the "Nurli uy" shop:

Reading the table: even if every twentieth lamp in the batch (5%) is faulty, the probability that all are fine in 20 checks is 36%. That is, you would miss such a "bad" batch in one case out of three.

How many should be checked? If you want to conclude with 95% confidence that "faulty lamps are below 5%", all n lamps must work, with 0.95ⁿ less than 5%:

n > ln(0.05) : ln(0.95) ≈ 58.4, that is, at least 59.

There is also a quick rule of thumb: if not a single faulty item is found in n checks, the 95% upper limit for the share of faulty items is approximately 3 : n. For 20 this is 3 : 20 = 15%. So your result is consistent with the conclusion "up to 15% may be faulty".

Practical tips:

1. Take the lamps to be checked at random from different places in the boxes: taking them from the top boxes distorts the result.
2. Agree the acceptance procedure with the supplier in the contract in advance: how many are checked and how many faulty ones mean the batch is returned.
3. Keep count of lamps returned by customers too: this is the cheapest and most continuous quality control.

In such cases a stratified sample is used. The population is divided into similar groups — strata — and a separate random sample is drawn from each stratum. The number of people taken from each stratum is made proportional to its share of the population (proportional allocation). Then the ratio of groups in the sample does not depend on chance.

Example (prepared by the author, hypothetical figures). The "Ohangaron qurilish mollari" shop has 2,000 customers: 1,400 retail, 400 wholesale, 200 corporate. 200 people are surveyed.

Overall satisfaction: (1,400 × 80% + 400 × 55% + 200 × 40%) : 2,000 = 71%.

Now say that in a simple random sample more retail customers happened to be included: 160 retail, 30 wholesale, 10 corporate. Even if satisfaction in each group is the same, the sample result comes out at (160 × 80% + 30 × 55% + 10 × 40%) : 200 = 74.25%, that is, 3.25 percentage points above the true figure. The reason is that the most satisfied group is over-represented in the sample.

An extra benefit of a stratified sample: you can draw a separate conclusion for each group. In this example it is immediately clear that the problem lies specifically with corporate customers.

If you deliberately survey more people from a small group (for example, 60 corporate customers to get a more precise picture), then when calculating the overall result each group is weighted by its actual share of the population, not by its number in the sample.

Caution is needed. The 300 respondents represent not all customers but only the part that subscribes to the channel and chose to answer the poll. In statistics this is called selection bias: the sample was not drawn from the population at random but "selected itself" by a certain trait. Such an error cannot be fixed by increasing the sample size: 3,000 answers instead of 300 would be skewed in the same direction.

In this poll, two things push the result upward:

1. Mostly loyal customers subscribe to the channel, while dissatisfied ones have often already left.
2. Answering the poll is voluntary: usually people with strong opinions answer, while indifferent customers stay silent.

Example (prepared by the author, hypothetical figures). "Lazzat oshxonasi" has 1,000 regular customers a month, 600 of whom subscribe to the channel. Say 85% of subscribers are satisfied and 50% of non-subscribers.

At best, the channel poll will show around 85%, and with self-selection among those willing to answer it can reach the 90% in the question. The true figure across all customers is 71%.

To get a more reliable result:

1. Draw the sample from the list of all customers (receipt database, loyalty card), not from channel subscribers.
2. Make the sample random: for example, ask every 10th customer a short question at the till, and do not let the staff member choose whom to ask.
3. Record the share of non-respondents. If it is large, interpret the result with caution.

The channel poll is not useless either: it shows the opinion of loyal customers well. Just do not announce it as "the opinion of all customers".

To compare indicators of different scale, a standardised deviation — the z-score — is convenient. It shows how many standard deviations the result has moved away from that branch's usual level:

z = (this month's result − usual average) : standard deviation

In this way each branch is measured against its own "usual fluctuation". If the distribution is roughly normal, about 95 percent of values lie within the average ± 2 standard deviations. So in practice, if the z-score is about 2 or more (or −2 or less), the result is considered unusual.

Example (prepared by the author, hypothetical figures). Based on data for the last 24 months:

In soums the large branch added more, but its revenue normally fluctuates by about ±16 million anyway, so +20 million is within its usual fluctuation. For the small branch, however, +12 million equals two standard deviations: for it this is a genuinely rare result, and its cause is worth investigating.

The z-score works on the negative side too: if in some month z comes out close to −2, this is not chance either but a signal to check.

Note: if seasonality is strong, take the average and standard deviation not for the whole year but for the same month in previous years. Otherwise December will come out "unusual" every year.

Cohort analysis is used for this. A cohort is a group of customers who came for the first time in the same period (for example, January customers). Then you track what percentage of each cohort came back in the following months:

Retention rate = those who came back in a later month : number of customers in the cohort

The total number of customers hides this picture: if newcomers replace those who leave, the total figure does not change and the problem is invisible.

Example (prepared by the author, hypothetical figures). The "Chiroyli soch" barbershop keeps track of customers by phone number.

How to read the table:

1. Compare cohorts of the same "age": after 1 month, 45 percent of January customers and 50 percent of February customers came back. Something may have improved in February (for example, reminder messages started being sent after the first visit).
2. Look at the curve within a row: the January cohort goes 45% → 35% → 30%. The decline is slowing, which means a "core" of regular customers of about 30 percent is forming.
3. The loss in the first month is the largest: 110 of the 200 did not come back. The biggest opportunity is right here — improving the first-visit experience.

Practical tip: update the cohort table every month by adding one new row and one new column. Raising retention by a few percentage points is often cheaper than attracting new customers.

The most common reason is that the vertical axis does not start at zero. In a bar chart the eye compares the heights of the bars. If the axis starts at, say, 495 million, only the part of each bar above 495 is drawn, and a small difference looks huge.

Example (prepared by the author, hypothetical figures). Monthly revenue: 500, 505, 512, 520 million soums.

Three things to check when you look at a chart:

1. Where the vertical axis starts. In a bar chart the axis must start at zero. In a line chart it may start above zero, but the axis labels must be clearly visible.
2. Whether the intervals on the horizontal axis are equal. If months are mixed with quarters in places, the growth rate is distorted.
3. Whether there are two different vertical axes. If revenue is on the left axis and profit on the right, the axes can be chosen to "bring the lines together" or "push them apart" at will.

The right way: if you want to show a small change, instead of bars plot the change itself (each month's percentage compared with the previous month) or write the exact figures next to the chart. Then the audience will see 4 percent as 4 percent.

This is not just a matter of good manners: a decision made on the basis of a misleading chart (for example, "we are growing fast, let's hire more staff") brings real costs to the company.

ABC analysis (ranking based on the Pareto principle) is handy for this. The steps are simple:

1. Write down the revenue of each product group for the period.
2. Sort them from largest to smallest.
3. Calculate each group's share of total revenue and the running (cumulative) share.
4. Split the groups into three classes by cumulative share.

Boundaries often used in practice: class A — up to roughly 80 percent of the cumulative share, B — the next roughly 15 percent, C — the rest. This is not a strict rule; the business chooses the boundaries itself.

Example (compiled by the author, hypothetical figures). Annual revenue of the "Nurafshon Household Goods" shop, mln soum:

Result: 4 groups out of 10 (40 percent of the range) bring in 80 percent of revenue. So class A goods must never run out, should stand in a prominent place on the shelves, and their prices should be monitored every week. For class C, stock can be reduced and orders placed less often.

Tip: run ABC analysis not only by revenue but also by gross profit. A product with large revenue but a small markup may fall into class B or C by profit — putting both tables side by side shows more precisely which product needs more attention.

The average price changes for two reasons: the price of each product changed, and the sales mix (how much of which product was sold) changed. In statistics these are separated with three indices:

• variable-composition index — current average price : base average price;
• fixed-composition index — revenue at current prices and current quantities : revenue at base prices and current quantities (pure price effect);
• structural shift index — average price at base prices and current mix : base average price (pure mix effect).

The first equals the product of the other two.

Example (compiled by the author, hypothetical figures). The "Tandirchi" bakery sells 1,000 loaves a day.

Calculation:

Check: 1.071 × 1.346 = 1.442.

Conclusion: the price rise lifted the average price by only 7.1 percent; the main growth (34.6 percent) came from customers buying more of the more expensive patir. This is an important distinction: if next month customers go back to ordinary bread, the average price will drop sharply even though prices have not changed, and this could be wrongly read as "prices fell".

Tip: when the "average receipt" or "average price" in the till report changes, check the mix as well before drawing conclusions — otherwise a wrong pricing decision will be made.

Comparing the largest and smallest salary looks at only two people and ignores those in between. Two methods are commonly used to show the whole distribution.

1. Funds ratio: total salary of the highest-paid group of employees : total salary of an equally sized lowest-paid group. In statistical practice it is often calculated by 10-percent groups (deciles). In a small team a 10-percent group comes down to one person and the ratio turns into a simple "highest : lowest" ratio, so here 20-percent groups (quintiles) are taken — this is called the quintile funds ratio.

1. The Lorenz curve and the Gini coefficient. Employees are split into equal groups in ascending order of salary, then the "cumulative share of employees" is compared with the "cumulative share of the payroll fund". The Gini coefficient ranges from 0 to 1: 0 — everyone earns the same; the closer to 1, the stronger the inequality. Formula for grouped data: G = 1 − Σ (Xₖ − Xₖ₋₁) × (Yₖ + Yₖ₋₁), where X is the cumulative share of employees and Y the cumulative share of salary.

Example (compiled by the author, hypothetical figures). The "Yashil Vodiy" workshop has 10 employees with monthly salaries (mln soum): 3, 3, 4, 4, 5, 5, 6, 6, 8, 16. The total fund is 60 mln soum. Each group has 2 people (20 percent):

Quintile funds ratio: 24 : 6 = 4. The ratio of the highest to the lowest salary, by contrast, is 16 : 3 ≈ 5.3 — it exaggerates the influence of a single employee (16 mln).

Gini coefficient: G = 1 − 0.2 × [(0.100 + 0) + (0.233 + 0.100) + (0.400 + 0.233) + (0.600 + 0.400) + (1.000 + 0.600)] = 1 − 0.2 × 3.667 ≈ 0.267. If each employee is taken separately, the exact value comes out at 0.28: grouping hides differences within groups and therefore lowers the coefficient slightly.

Practical benefit: if you calculate the coefficient the same way every year, you will clearly see whether changes in the pay system (such as a new bonus scheme) increased or reduced inequality. Be careful when comparing one firm's coefficient with another firm's or a country's figure: if the number of employees and the grouping method differ, the comparison will not be valid.

A share obtained from a sample is always an estimate. Its accuracy can be shown by the margin of error. A simple formula for a confidence level of about 95 percent:

margin of error ≈ 1.96 × √(p × (1 − p) : n),

where p is the share in the sample and n is the sample size. This formula works only if the sample is truly random, and it gives a good result when the sample is small relative to the whole population.

Example (compiled by the author, hypothetical figures). "Hilol Optika" has 2,000 customers in its database; of 200 randomly selected people, 120 are satisfied.

So rather than saying "60 percent are satisfied", it is more honest to say "roughly 53 to 67 percent". If next quarter's survey shows 63 percent, that increase is within the margin of error — it is too early to announce it as an "improvement".

Required sample size. If you want the error not to exceed ±5 percentage points, for the most cautious case (p = 0.5): n = 1.96² × 0.5 × 0.5 : 0.05² ≈ 384.2, i.e. 385 people. Since the population is only 2,000 people, the finite population correction is applied: n = 384.16 : (1 + 383.16 : 2,000) ≈ 322.4, i.e. 323 people when rounded up.

Bear in mind: enlarging the sample only reduces random error. If the survey is carried out only among customers who came into the shop themselves, or is sent only to regular customers, the result is skewed to one side, and no formula will correct that.

Comparing one month with the same month of last year removes seasonality, but it is very sensitive to random factors (holidays, weather, days closed for repairs). So alongside it you should compare cumulative revenue since the start of the year with the same period of last year. The third useful indicator is the series of growth rates of each month against the same month of last year: it shows in advance that growth is slowing.

Example (compiled by the author, hypothetical figures). Monthly revenue of the "Bahor" cafe, mln soum:

Three conclusions follow:

1. Since the start of the year the cafe has earned 10 percent more revenue than last year — the overall picture is not bad.
2. The October drop is no accident: the growth rate has been falling every month since July (9.1 → 7.8 → 4.4 → −8.0). In other words, the problem did not arise in one month but built up over several months.
3. Looking only at the cumulative indicator, this slowdown would have gone unnoticed; looking only at October, the alarm would have been exaggerated.

Tip: include all three indicators — the current month, year to date and the series of growth rates — in one table in the monthly report. If the slowdown continues for 2–3 months in a row, that is exactly the point from which to start looking for the cause.

If revenue is growing steadily, there are two simple methods: the average absolute increase and the average growth rate. The first assumes an increase of roughly the same amount each month, the second an increase of the same percentage.

Example (prepared by the author, hypothetical figures). Revenue of the "Guliston qurilish mollari" shop, million soums:

Method 1. Average absolute increase = (last − first) : (number of periods − 1) = (55 − 40) : 5 = 3 million soums. July estimate: 55 + 3 = 58 million soums.

Method 2. Average growth rate = 5th root of (55 : 40) ≈ 1.066, i.e. about 6.6% a month. July estimate: 55 × 1.066 ≈ 58.6 million soums.

The two results are close, so the purchasing plan can be drawn up for sales of 58–59 million soums.

Limitations:

• these methods are suitable only for estimating one or two months ahead;
• if revenue depends on the season (for example, building materials sell less in winter), an estimate made without taking seasonality into account will be wrong;
• if the first or last month is unusual, the result is distorted, because the formula relies only on these two points.

It is convenient to take revenue data from the monthly credit turnover of account 9020 "Revenue from sale of goods".

Quartiles are convenient for this. The values are arranged in ascending order and divided into four equal parts: Q1 (the first quartile) — a quarter of the values are smaller than it; Q2 — the median; Q3 — three quarters of the values are smaller than it. The range from Q1 to Q3 contains the middle half of the receipts, and a single large receipt has almost no effect on it.

Example (prepared by the author, hypothetical figures). "Bahor kafe" had 12 receipts in one day, thousand soums: 18, 22, 25, 27, 30, 32, 35, 38, 41, 45, 60, 140.

Conclusion: a typical receipt is in the range of 26–43 thousand soums. The 140-thousand-soum receipt is well above the upper limit (68.5), i.e. it is an outlier: probably a banquet or a large order. It has pushed the arithmetic mean up to 42.75, whereas half of the receipts are smaller than 33.5.

The "Q3 + 1.5 × range" rule is a widespread method for finding outliers. Also remember that there are several methods of calculating quartiles: spreadsheet programs may give a slightly different result, so stick to one method within one analysis.

Practical benefit: you can set the price of a combo meal to fit the typical range and analyse large orders separately.

How much two indicators change together is shown by the Pearson correlation coefficient (r). It ranges from −1 to +1: close to +1 — when one rises, the other rises steadily too; close to 0 — there is no linear relationship; close to −1 — when one rises, the other falls.

Formula: r = Σ(dx × dy) : √(Σdx² × Σdy²), where dx and dy are the deviations of each value from its own mean.

Example (prepared by the author, hypothetical figures). "Navbahor mebel" over 5 months, million soums:

r = 32 : √(10 × 114) = 32 : 33.76 ≈ 0.95. This is a very strong positive relationship. r² ≈ 0.90, i.e. about 90% of the variation in revenue is explained through the linear relationship with advertising (in the statistical sense, not as a cause). The slope is 32 : 10 = 3.2: in months when advertising was 1 million soums higher, revenue was on average 3.2 million soums higher.

Be careful:

• a strong correlation does not prove cause and effect: if advertising was increased during the season, it may have been the season that raised revenue;
• 5 observations are very few; check the conclusion with at least a year of data.

It is easy to get the amount of advertising expense from the accounts: advertising and marketing services received from outside are recorded with Dt 9410 "Selling expenses" – Ct 6010 (NAS No. 21, paragraph 412, line 2).

This is called Simpson's paradox: even if one side is ahead in each of the groups, when the groups are combined the result may come out the other way round. The reason is that the groups are of different sizes: if someone got many customers from an easy group, his overall percentage comes out high.

Example (prepared by the author, hypothetical figures). The indicator is what percentage of the customers approached made a purchase.

Anvar is better in both groups. But Botir's customers are mostly retail (250), where the purchase rate is high anyway. Half of Anvar's customers, on the other hand, are from the wholesale group, which is hard to sell to. The overall percentage reflects this difference in composition, not skill.

For a correct comparison:

• look at the results by group separately;
• or calculate by applying the same customer mix to both managers: for example, with 100 from each group, Anvar (80 + 20) : 200 = 50%, Botir (72 + 10) : 200 = 41%.

Conclusion: when rewarding managers, do not look only at the overall percentage — also take into account which customers were assigned to whom.

Because months differ in length, comparing monthly totals directly can lead to the wrong conclusion. First bring the indicator to a per-day basis (for a business that works every day) or a per-working-day basis (for a business closed on days off), then compare.

Example (prepared by the author, hypothetical figures). "Samarqand non uyi" works every day:

Calculation: 171.6 : 186 = 0.923 (the monthly total is 7.7% lower); 171.6 : 28 = 6.129 and 6.129 : 6 = 1.021 (daily revenue is 2.1% higher). So trade has not actually worsened, but has improved slightly.

Take two more things into account:

• days off and holidays: if there are closed days, divide by the number of days worked, not calendar days;
• the season: comparing February not with January but with February of the previous year also removes the effect of the season.

In the report, give both indicators side by side: the monthly total is needed for comparison with the plan, and the daily average for assessing the pace of work.

A balance is a state at a single moment (a point-in-time indicator). For such a series, not the simple arithmetic mean but the chronological mean is used: half of the first and last balances and the full intermediate balances are taken, and the sum is divided by the number of intervals (the number of dates minus one). The reason: each month lies between two dates, while the end dates "serve" only one month each.

Example (compiled by the author, hypothetical figures). Balance of 2910 "Goods in the warehouse" for the "Yulduzcha" shop:

Chronological mean: (40 : 2 + 46 + 50 + 44 : 2) : 3 = (20 + 46 + 50 + 22) : 3 = 138 : 3 = 46 million soums.

If we simply add the four balances and divide by 4, we get (40 + 46 + 50 + 44) : 4 = 45 million soums — this gives excess weight to the beginning and end of the quarter.

Tip: average inventory goes into the denominator of indicators such as inventory turnover. So calculate it the same way each time; otherwise comparing different quarters will be misleading.

No, in this case the simple arithmetic mean overstates the average price. Because the amount is the same each time, more sacks are bought when the price is low and fewer when it is high. For such a series the harmonic mean is used: the number of values is divided by the sum of their reciprocals.

Harmonic mean = n : (1/x₁ + 1/x₂ + … + 1/xₙ).

In practice this gives the same result as "total money spent : total quantity obtained".

Example (compiled by the author, hypothetical figures). The "Tandir non" bakery bought flour three times during the month, for 12,000,000 soums each time.

Correct average price: 36,000,000 : 120 = 300,000 soums. Check with the harmonic formula: 3 : (1/240,000 + 1/300,000 + 1/400,000) = 3 : (12/1,200,000) = 300,000 soums.

Simple arithmetic mean: (240,000 + 300,000 + 400,000) : 3 = 313,333 soums. This is 13,333 soums too high, because it gives the same "weight" to the 50 sacks bought at the low price and the 30 sacks bought at the high price.

Which mean to choose depends on what data you have:

• if the quantity bought at each price is known — the arithmetic mean weighted by quantity;
• if the amount spent at each price is known — the harmonic mean (if the amounts are not equal, the weighted harmonic mean: total amount : sum of each amount divided by its price).

Link with accounting: if there was no opening balance of flour at the beginning of the month, under the AVECO method the cost of one sack also comes out at exactly 300,000 soums, because under paragraph 65 of NAS No. 4 the weighted average cost is determined from total cost and total quantity.

No, comparing objects of different sizes by absolute amounts is misleading. For this, a relative intensity indicator is used: one indicator is divided by another, related indicator — for example, revenue by sales floor area or by number of staff. You take the revenue amount for the month from the credit turnover of the income account.

Example (compiled by the author, hypothetical figures). November:

Calculation: 90,000,000 : 120 = 750,000; 54,000,000 : 45 = 1,200,000; 90,000,000 : 6 = 15,000,000; 54,000,000 : 3 = 18,000,000.

The large shop has more revenue, but each square metre and each salesperson earns more money in the small shop. This may point to idle space or slow-moving goods in the large shop — the reason needs to be analysed separately.

Revenue can be viewed as the product of two factors: revenue = number of receipts × average receipt. You separate the effect of each factor using the chain substitution method: first the quantitative factor (number of receipts) changes while the qualitative factor (average receipt) stays at the old level; then the qualitative factor is also replaced with the new one.

Example (compiled by the author, hypothetical figures). The "Qaldirg'och" café:

Total growth: 143,000,000 − 120,000,000 = 23,000,000 soums.

• Effect of the number of receipts: (2,200 − 2,000) × 60,000 = 12,000,000 soums;
• effect of the average receipt: 2,200 × (65,000 − 60,000) = 11,000,000 soums.

Check: 12,000,000 + 11,000,000 = 23,000,000 soums.

Conclusion: almost half of the growth came from new customers, the rest from larger orders. If the average receipt grew only because prices rose, check this separately too — the number of customers may fall in the following months.

This is shown by the absolute value of one percent of growth: the absolute growth is divided by the growth rate (in percent). The result is always equal to one hundredth of the previous period's level. That is, the same percentage growth on a larger base means more soums.

Example (compiled by the author, hypothetical figures):

Calculation: 8 : 10 = 0.8 million; 3 : 10 = 0.3 million. Check: 80 : 100 = 0.8; 30 : 100 = 0.3.

Practical benefit: a 20% growth plan for the small branch brings in even less additional money than 10% at "Shimol" (30 × 20% = 6 million, 80 × 10% = 8 million). That is why targets should be set not only in percent but also in soums.

The arithmetic mean is sensitive to a single very large value: the director's high salary pulls the average up. In such a case the salary of a "typical" employee is better shown by the median: the number in the middle when the values are sorted.

With an even number of values, the median is the average of the two middle numbers. That is why it is useful to report both together: a large difference means there is a sharply outlying value in the distribution.

Example (prepared by the author, hypothetical figures): at "Mehr dizayn" 6 people earn per month (mln soum): 3; 3.5; 4; 4.5; 5; 25. The sum is 45, the average 45 / 6 = 7.5 mln. Median (4 + 4.5) / 2 = 4.25 mln. Five out of six earn less than 7.5 mln, while the median shows their real level.

No. Percentages are calculated on a base that changes every year, so simply adding them up and dividing gives a wrong result. The average annual growth rate is found with the geometric mean: the annual ratios are multiplied together and the root of the power equal to the number of years is taken.

It is easy to check: if you apply the average rate you found to each year, you should get the actual result for the last year. A simple average fails this test.

Example (prepared by the author, hypothetical figures): sales of "Durdona kitob" were 100 mln soum. Year 1: 100 × 1.3 = 130 mln. Year 2: 130 × 0.9 = 117 mln. The product of the ratios is 1.3 × 0.9 = 1.17; the root √1.17 ≈ 1.0817, that is, average annual growth ≈ 8.17%. Check: 100 × 1.0817 × 1.0817 ≈ 117 mln. With "10%", however, the result would be 121 mln, which is wrong.

A moving average is used for this. Each time, the average of several consecutive periods is taken, and then the window is shifted one step forward. Random jumps are smoothed out and the underlying trend becomes visible.

The wider the window, the smoother the line, but the later it responds to change. For ordinary sales, a window of 3 or 4 periods is often used.

Example (prepared by the author, hypothetical figures): sales of "Yulduz nonvoyxonasi" over 6 weeks (mln soum): 10, 14, 9, 13, 16, 12. Three-week moving average: (10 + 14 + 9) / 3 = 11; (14 + 9 + 13) / 3 = 12; (9 + 13 + 16) / 3 ≈ 12.67; (13 + 16 + 12) / 3 ≈ 13.67. Even though the weekly figures jump around, the averages rise steadily from 11 to 13.67.

The seasonality index is the ratio of the figure for each period to the average of all periods, as a percentage. If it is above 100%, that period is stronger than average; if below, weaker. The average of the indices of all quarters is 100%.

This index helps in planning purchases and staffing. For a more accurate result, it is advisable to take several years of data and average each quarter across the years.

Example (prepared by the author, hypothetical figures): quarterly sales of the "Muzbek" workshop (mln soum): 20, 60, 90, 30. Quarterly average: 200 / 4 = 50. Indices: 20 / 50 = 40%; 60 / 50 = 120%; 90 / 50 = 180%; 30 / 50 = 60%. Check: (40 + 120 + 180 + 60) / 4 = 100%. So in the 3rd quarter 1.8 times more stock is needed than in an average quarter.

A structure indicator is the ratio of each part to the whole, as a percentage. The amount of each group is divided by the total and multiplied by 100. The sum of all shares must be 100%; this is the simplest check.

Comparing the structure across periods is especially useful: even if total revenue does not change, a shift in shares shows that customers' tastes are changing. A change in a share is expressed in percentage points.

Example (prepared by the author, hypothetical figures): the monthly revenue of "Kafe Bodom" is 50 mln soum: food 30, drinks 12, desserts 8. Shares: 30 / 50 = 60%; 12 / 50 = 24%; 8 / 50 = 16%; total 100%. If next month the dessert share rises from 16% to 20%, this is a 4 percentage point increase: a signal to expand the menu.

For this, a "basket" price index is calculated. Quantities are fixed as in the base period, and then the value of the basket at current prices is divided by its value at base prices. Since the quantities do not change, the index shows only the effect of prices.

It is not the price rise of a single product, but the products with a large share in the basket, that affect the index most. That is why the index cannot be replaced with a simple average of each product's price rise.

Example (prepared by the author, hypothetical figures): the basket: 4 loaves of bread and 2 litres of milk. Base prices: bread 3 000, milk 10 000 soum. Current: bread 3 300, milk 12 000. Base basket: 4 × 3 000 + 2 × 10 000 = 32 000. Current basket: 4 × 3 300 + 2 × 12 000 = 37 200. Index: 37 200 / 32 000 = 1.1625, that is, prices rose by 16.25%. A simple average of the product price rises, by contrast, would be (10% + 20%) / 2 = 15%.

Growth measured in money (nominal growth) contains two things: selling more and selling at higher prices. To find the actual (real) growth, current revenue is divided by the price index, that is, it is restated at base prices. This operation is called deflating.

Note: subtracting 10% from 15% (5%) gives an approximate result. The correct calculation is by division: 1.15 / 1.10.

Example (prepared by the author, hypothetical figures): revenue of the "Sardor maishiy" shop rose from 200 mln soum to 230 mln soum, and prices rose by 10%. Revenue at base prices: 230 / 1.1 ≈ 209.09 mln. Real growth: 209.09 / 200 − 1 ≈ 4.55%. That is, the volume of goods sold increased by about 4.55%, and the rest is due to prices.

No. Two indicators changing together (association, correlation) does not prove that one causes the other. Often there is a third factor affecting both.

In this example, the third factor is rainy, cool weather. It increases demand both for umbrellas and for hot tea. If you make a decision without knowing the cause, for example, spend money on advertising umbrellas, tea sales will not rise.

Example (prepared by the author, hypothetical figures): the 10-day log of "Oqsaroy do'koni": on 5 dry days, on average 2 umbrellas and 30 cups of tea were sold; on 5 rainy days, 12 umbrellas and 70 cups of tea. The association looks strong, but if you split the days by weather, it becomes clear that both sales follow the weather.

A properly designed sample gives a good estimate of the whole population. The key condition: every customer must have an equal chance of being included in the sample (random selection). Then the share in the sample comes out close to the share in the whole population.

The error often lies in the selection method: surveying only morning customers or only regular customers skews the result to one side. Increasing the sample size does not correct this error. Also remember that the result is an estimate and has a certain margin of error.

Example (prepared by the author, hypothetical figures): "Hilol optika" randomly selected 200 people from its database of 2 000 customers and surveyed them. 120 people are satisfied with the service: 120 / 200 = 60%. Estimate for all 2 000 customers: around 2 000 × 60% = 1 200 satisfied customers.

For this, a grouped frequency table is prepared. First the smallest and largest values are identified, and then intervals of equal width are chosen. So that each value falls into only one interval, the boundary rule is stated in advance: for example, the interval "from 50 to 100" includes 50 but not 100.

For each interval, the count (frequency) and the share are recorded. The sum of the frequencies must equal the total number of observations, and the shares must add up to 100%.

Example (prepared by the author, hypothetical figures): receipts at "Anor savdo" (thousand soum): 35, 48, 52, 61, 74, 79, 83, 95, 102, 118, 125, 142. Intervals: 0–50: 2 (≈16.7%); 50–100: 6 (50%); 100–150: 4 (≈33.3%). Total 12, 100%. Clearly, half of the purchases are in the range of 50–100 thousand soum.

The mode is the value that occurs most often in a data set. The difference from the arithmetic mean: it is not calculated but found by counting, and it is always a value that actually exists.

Example: the sizes of pairs sold in one day at the "Qadam" shoe shop: 39, 41, 42, 41, 40, 41, 43, 42, 41, 38. We count: size 41 appears 4 times, size 42 twice, and the rest once each. Mode = 41. The arithmetic mean comes out at 40.8, and there is no such size.

Practical use: when placing orders, you need the mode, not the average. The shop should stock more size 41. If two values occur equally often, the data set has two modes.

Here a simple arithmetic mean gives the wrong result, because the dishes were sold in different quantities. The correct method is the weighted average: each price is multiplied by the number sold, and the total is divided by the total quantity.

Example: the "Tandir" café: shurpa 25 000 soum × 40 portions = 1 000 000; plov 35 000 soum × 100 portions = 3 500 000; kebab 60 000 soum × 20 portions = 1 200 000. Total: 5 700 000 soum, 160 portions. Weighted average: 5 700 000 ÷ 160 = 35 625 soum.

A simple average would be (25 000 + 35 000 + 60 000) ÷ 3 = 40 000 soum and would overstate the café's revenue. The weighted average cost (AVECO) method in accounting is built on exactly the same logic.

An average does not show fluctuation. The simplest measure of fluctuation is the range: the largest value − the smallest value. The smaller the range, the more stable the indicator.

Example: revenue of the two "Sarbon market" branches over 5 days (mln soum): Branch 1: 9, 10, 10, 11, 10. Average 10, range 11 − 9 = 2. Branch 2: 4, 16, 7, 15, 8. Average 10, range 16 − 4 = 12. The averages are the same, but at branch 2 revenue jumps sharply.

Practical conclusion: at branch 2 it is harder to plan cashiers and stock, so the reasons (market days, promotions, cash desk discipline) need to be studied separately. The range looks only at the two extreme values, so a single unusual day can sharply increase it.

The first is the chain growth rate, the second is the base growth rate. In the chain method, each month is divided by the month before it. In the base method, each month is divided by one chosen base month (for example, January).

Example: revenue of "Mehr optika" (mln soum): January 50, February 55, March 66. Chain: February 55 ÷ 50 = 110%, March 66 ÷ 55 = 120%. Base (January = 100%): February 110%, March 66 ÷ 50 = 132%.

The relationship: the product of the chain ratios equals the base ratio: 1.10 × 1.20 = 1.32. The chain rate shows month-to-month acceleration or slowdown, while the base rate shows the overall result since the start of the period.

More precisely, it increased by 2 percentage points. A percentage point is the simple difference between two percentage figures: 12% − 10% = 2 p.p. The relative change is something else: (12 − 10) ÷ 10 × 100% = 20%. That is, the indicator rose by 20 percent relative to its own level.

Mixing the two up causes serious confusion in reports: if you say "increased by 2 percent", the reader may understand 10.2%.

Example: at the "Oltin don" flour mill, the share of returned products rose from 4% to 6%. The correct wording: "the return share rose by 2 percentage points, which is a 50 percent increase in relative terms". For the manager, the second figure better shows the scale of the problem.