Statistical Inference · Statistical Inference
Why samples disagree
In Statistical Inference because one week of the shop is noisy — a second eight customers at the same counter already move the numbers.
Week 1's mean spent is Rs 220, and 5 of 8 came back. That is not a law of the shop. Next week eight different people walk in. Their names are Isha, Jai, Kiran, Lata, Mira, Naveen, Omar, Priya. Same columns — visits, spent, plan, came_back — new rows. Isha spent Rs 180; Priya spent Rs 550. If the numbers move, that is sampling variation, not a new business.
- Statistical Inference
- Medium level
- 5 concepts
1A second eight customers
Week 2 uses the same five columns. Isha spent Rs 180, Jai Rs 90, Kiran Rs 420, Lata Rs 120, Mira Rs 300, Naveen Rs 0, Omar Rs 140, Priya Rs 550. Add them: 180 + 90 + 420 + 120 + 300 + 0 + 140 + 550 = 1800. Divide by eight: 1800 / 8 = 225. Mean spent moved from 220 to 225.
Came_back is again 5 yes and 3 no. Visits sum to 24, so mean visits is 24 / 8 = 3 instead of week 1's 22 / 8 = 2.75. Same shop, same question, new people. The sheet changed because the sample changed — not because the counter was rebuilt.
No diagram — the idea is carried by the prose, table, code block or coding lab.
| name | visits | spent | plan | came_back |
|---|---|---|---|---|
| Isha | 3 | 180 | monthly | yes |
| Jai | 1 | 90 | drop-in | no |
| Kiran | 5 | 420 | monthly | yes |
| Lata | 2 | 120 | drop-in | no |
| Mira | 4 | 300 | monthly | yes |
| Naveen | 0 | 0 | monthly | no |
| Omar | 2 | 140 | drop-in | yes |
| Priya | 7 | 550 | monthly | yes |
Week-2 mean spent
What is mean spent on the eight week-2 rows?
- 180+90+420+120+300+0+140+5501800
- 1800 / 8225
Pro tip. Week 1 was 220. The shop did not hire a new statistician. The customers changed.
Coding lab. Week-2 mean runs in the app, with checks on your output.
Week 1 mean spend is Rs 220 and Week 2 mean spend is Rs 225 across different customer samples from the same shop. What explains this difference?
- Natural sampling variation arising from drawing different random individuals from the same population
- A systemic shift in the underlying economic pricing structure of the store
- A measurement calculation error in one of the two spreadsheet columns
- Deterministic proof that the shop customer base expanded between weeks
Different random samples from the same steady population naturally yield slightly different sample statistics.
2Same counter, different people
Nothing about the shop's typical customer had to change for 220 to become 225. Week 2 is not a redesign and not a new product. It is eight other people who happened to walk in: Isha instead of Anu, Priya instead of Chitra.
If you treat week 1 as the shop, you will panic when week 2 disagrees by five rupees. The honest reading is: each week is one sample from the same population. Disagreement is what samples do. A five-rupee move on a mean of about 220 is a new draw, not a new business.
Figure. Week-1 mean spent is 220. Week-2 mean spent is 225. Same shop, two samples.
The means moved by five rupees
How far apart are the two week means?
- week-2 mean225
- 225 - 2205
Pro tip. Five rupees on a 220 week is a wobble, not a new shop.
Coding lab. Two means runs in the app, with checks on your output.
Why does sample mean fluctuate between independent batches of customers even when the shop remains unchanged?
- Because the shop true mean changes value every time a sample is collected
- Because pandas applies a random scaling factor to column calculations
- Because sample variance is forced to zero during data ingestion
- Because each sample captures a different combination of high and low spenders by chance
Random selection captures different subsets of customers, producing sampling wobble around the population mean.
3One week is noisy
Eight people is a small sample. Chitra's Rs 510 and Deepak's Rs 0 both sit in week 1. Swap Chitra for someone who spent Rs 150 and the week total drops by 360, so the mean drops by 360 / 8 = 45. That sensitivity is why one week is noisy: one loud spender is an eighth of the story.
Week 2's Rs 5 shift is ordinary noise, not evidence the shop changed. Inference will later measure that noise as a standard error. For now, expect the next eight customers to disagree a little — and sometimes a lot — even when the typical customer stayed put.
No diagram — the idea is carried by the prose, table, code block or coding lab.
| Number | Week 1 | Week 2 |
|---|---|---|
| Mean spent | 220 | 225 |
| Mean visits | 2.75 | 3 |
| Came back | 5/8 | 5/8 |
Coding lab. Drop Chitra runs in the app, with checks on your output.
Week 2 mean spent is Rs 225 after week 1 was Rs 220. The honest reading is:
- Sampling variation — a new eight people at the same counter
- Proof the shop raised prices
- A reason to fit a neural net
Different samples, different numbers. The counter did not become a new business.
4More rows, less wobble
If the owner kept thirty-two customers instead of eight, one loud spender would tug the mean less. On eight rows, Chitra is 1 / 8 of the mean. On thirty-two rows she is 1 / 32. The week total is divided by a larger n — n is just the number of people in the sample — so each person is a smaller share.
That is not a promise of a perfect number. It is a promise of a calmer one. Inference will write this as standard error later: the wobble shrinks like 1 over the square root of n, not like 1 over n. The plain version: a bigger sample wobbles less from week to week.
Figure. On eight rows, Chitra's Rs 510 is 1/8 of the mean — 510 of 1760. On thirty-two rows the same 510 is 1/32. The two bars are those shares, to scale with each other (4 : 1). A bigger n does not promise a perfect number; it promises a calmer one.
Why n calms the mean
- Small nOne Chitra is 510 of 1760 — a large share.
- Larger nThe same 510 is a smaller share of a longer list.
- Calmer meanNext week's mean sits closer to this week's.
One loud spender's share
Chitra spent Rs 510. What share of the week mean is she when n is 8, and when n is 32?
- share of the mean when n = 81 / 8 = 0.125
- share of the mean when n = 321 / 32 = 0.03125
Pro tip. Four times the rows quarters her share of the mean. The later SE formula shrinks more slowly than that, because it uses a square root.
Coding lab. Stack two weeks runs in the app, with checks on your output.
What happens to the variability of the sample mean when you increase sample size from 8 to 32 customers?
- The sample mean becomes less volatile because extreme individual spenders exert less pull on the average
- The sample mean fluctuates four times more wildly around the truth
- The population standard deviation shrinks to zero immediately
- The sample mean is guaranteed to match the population mean with zero error
Larger samples average out individual extremes, reducing sampling variance and yielding more stable estimates.
5Do not crown one week
A single sheet is one draw. Crowning week 1 as the shop makes week 2's 225 look like a scandal. It is not. It is another draw from the same counter, with Isha through Priya instead of Anu through Hari.
The cure is not to ignore the sheet. The cure is to remember it is a sample, then measure how much samples like it wobble — the next lesson's pile of would-repeat means. Read 220 as this week's statistic, not as the shop.
Figure. Each week is one sample. Week 1 is 220, week 2 is 225. Do not crown week 1 as the shop just because it arrived first.
| Move | Honest? |
|---|---|
| Treat week 1 as one sample | Yes |
| Crown week 1 as the shop | No |
Coding lab. Print both stories runs in the app, with checks on your output.
Why is treating a single week sample mean of Rs 220 as the permanent truth of the shop a mistake?
- Because sample means never bear any mathematical relation to population parameters
- Because customer spend amounts must follow a uniform distribution between 0 and 100
- A single sample is one random draw from a distribution and carries sample-specific wobble
- Because the true population mean can only be an even integer number
Overinterpreting one draw ignores sampling error; any single sample is just one point from the sampling distribution.
Notes
- In Statistical Inference because one week of the shop is noisy — a second eight customers at the same counter already move the numbers.
- Week 2 uses the same five columns. Isha spent Rs 180, Jai Rs 90, Kiran Rs 420, Lata Rs 120, Mira Rs 300, Naveen Rs 0, Omar Rs 140, Priya Rs 550. Add them: 1800. Divide by eight: 225. Mean spent moved from 220 to 225.
- Nothing about the shop's typical customer had to change for 220 to become 225. Week 2 is not a redesign. It is eight other people who happened to walk in.
Exam traps & shortcuts
- A sample number describes these rows. A population claim reaches past them.
- Fail-to-reject is not proof. A p-value is not the chance the null is true.
Recap
Two weeks, two means. 220 and 225 are both honest. One week is noisy; a bigger sample calms the wobble. Do not over-read a single sheet.
- A second eight customers
- Week 2 uses the same five columns. Isha spent Rs 180, Jai Rs 90, Kiran Rs 420, Lata Rs 120, Mira Rs 300, Naveen Rs 0, Omar Rs 140, Priya Rs 550. Add them: 180 + 90 + 420 + 120 + 300 + 0 + 140 + 550 = 1800. Divide by eight: 1800 / 8 = 225. Mean spent moved from 220 to 225.
- Same counter, different people
- Nothing about the shop's typical customer had to change for 220 to become 225. Week 2 is not a redesign and not a new product. It is eight other people who happened to walk in: Isha instead of Anu, Priya instead of Chitra.
- One week is noisy
- Eight people is a small sample. Chitra's Rs 510 and Deepak's Rs 0 both sit in week 1. Swap Chitra for someone who spent Rs 150 and the week total drops by 360, so the mean drops by 360 / 8 = 45. That sensitivity is why one week is noisy: one loud spender is an eighth of the story.
- More rows, less wobble
- If the owner kept thirty-two customers instead of eight, one loud spender would tug the mean less. On eight rows, Chitra is 1 / 8 of the mean. On thirty-two rows she is 1 / 32. The week total is divided by a larger n — n is just the number of people in the sample — so each person is a smaller share.
Practise Why samples disagree
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- A 2-question practice set that ends the chapter
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- Timed mocks scored with the real marking scheme
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