Range, variance and standard deviation explained

You

Alex, help me work through “Spread: range, SD and quartiles” step by step.

Alex Tutor

CatChow works with two courier companies. Customers are promised delivery within 36 hours, and the operations lead wants to drop one courier to simplify contracts. She pulls the last 10 deliveries for each and reports: "Both average exactly 24 hours. It doesn't matter which one we keep."

It matters a lot. Averages tell you where the data is centred, but nothing about how far individual values stray from that centre. A courier who always takes 23–26 hours and one who takes anywhere from 14 to 38 hours have the same mean, but only one of them keeps the 36-hour promise every time.

This lesson is about measures of spread (also called variability): range, variance, standard deviation, quartiles and the interquartile range, plus the coefficient of variation for comparing spread across different metrics. Download the deliveries and follow along.

Alex Tutor

Dataset · 20 rows

CatChow deliveries by courier

The last 10 deliveries by each of two couriers. Synthetic data made for this course.

Download CSV

Columns
delivery_id: Delivery number
courier: Courier company: A or B
hours: Hours from dispatch to delivery

Data

delivery_id	courier	hours
D-301	A	24
D-302	B	24
D-303	A	24
D-304	A	21
D-305	B	14

Show 15 more rows

D-306	B	18
D-307	A	23
D-308	A	21
D-309	B	30
D-310	A	26
D-311	B	14
D-312	A	26
D-313	A	26
D-314	B	18
D-315	A	23
D-316	A	26
D-317	B	30
D-318	B	38
D-319	B	30
D-320	B	24

Alex Tutor

Delivery times: courier A vs courier BBoth couriers are centred on 24 hours. Courier A's deliveries sit in a tight cluster; courier B's are spread from 14 to 38 hours, and one breaks the 36-hour promise.

Chart data

Delivery time, hours	Courier A	Courier B
12–15	0	2
16–19	0	2
20–23	4	0
24–27	6	2
28–31	0	3

Show 2 more rows

32–35	0	0
36–39	0	1

Alex Tutor

Question 1

Using the dataset CatChow deliveries by courier, compute the sample standard deviation of courier A's delivery times.

Your answer

Units: hours, e.g. 4.5

Alex Tutor

The range is the largest value minus the smallest. Courier A:

26 - 21 = 5

hours. Courier B:

38 - 14 = 24

hours. Already the difference is obvious.

The range is easy to compute and easy to explain, but it depends on just two values, the two most extreme ones. One lost parcel that took 90 hours would change courier A's range from 5 to 69, even if the other nine deliveries were perfect. Use the range as a first glance, not as the final word.

Alex Tutor

A better idea is to use every value. For each delivery, compute its deviation: the value minus the mean. For courier B, sorted, the deviations from 24 hours are:

Hours	14	14	18	18	24	24	30	30	30	38
Deviation	−10	−10	−6	−6	0	0	+6	+6	+6	+14

Why not simply average the deviations? Try it: they add up to exactly zero. That's not a coincidence. The mean is the balance point of the data, so the negative deviations always cancel the positive ones. To measure spread we need to get rid of the signs.

Alex Tutor

Question 2

In the same dataset, what is the interquartile range of courier A's delivery times?

Your answer

Units: hours, e.g. 4.5

Alex Tutor

\sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i - n\bar{x} = 0

Deviations from the mean always sum to zero, because n times the mean equals the total.

Alex Tutor

The standard trick is to square each deviation: squares are never negative, and big misses count much more than small ones (a deviation of 14 becomes 196). For courier B the squared deviations are 100, 100, 36, 36, 0, 0, 36, 36, 36 and 196, which add up to 576.

Now average them, with one twist. When your data is a sample (10 deliveries, not every delivery the courier will ever make) you divide by

n - 1

instead of

n

. Why? The sample mean is computed from the same 10 values, so it sits a little closer to them than the true long-run mean does. That makes the deviations look slightly too small, and dividing by

n - 1

compensates. So the sample variance is

576 / 9 = 64

.

If you truly have the whole population (say, every order ever placed, and you only care about those orders), divide by

n

:

576 / 10 = 57.6

. In product analytics you almost always work with samples, so

n - 1

is the default.

There is a catch: squaring also squares the units. Variance is 64 hours², and nobody can picture a squared hour.

Alex Tutor

Question 3

If delivery time is measured in hours, its variance is also measured in hours.

TrueFalse

Alex Tutor

s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2 \qquad \sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2

Left: sample variance (divide by n − 1). Right: population variance (divide by N, deviations from the population mean μ).

Alex Tutor

Take the square root of the variance and you are back in the original units. That's the standard deviation (SD):

\sqrt{64} = 8

hours for courier B. Read it as "a typical delivery is about 8 hours away from the average". Do the same for courier A and you'll get a much smaller number: that's your first quiz question.

For data that forms a roughly symmetric, bell-shaped hill, there is a handy rule of thumb: about 68% of values fall within one SD of the mean, and about 95% within two SDs. For courier B, "two SDs" is

24 \pm 16

, or 8 to 40 hours, and the 36-hour promise sits inside that band. Some late deliveries are to be expected, not a fluke. (Ten deliveries are too few to trust the percentages exactly, and in module 2 you'll see when this bell shape can be assumed.)

Always report spread next to the centre: "24 ± 8 hours" says far more than "24 hours on average".

Alex Tutor

Question 4

Put the steps for computing a sample standard deviation in the right order.

Tap an item, then tap where it should go. You can also drag or use the arrows.

Alex Tutor

s = \sqrt{s^2} = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}

Sample standard deviation: the square root of the sample variance, in the same units as the data.

Alex Tutor

Question 5

One parcel gets lost and arrives after 90 hours. Which measures of courier A's spread would change a lot? Select all that apply.

Choose all that applyRangeStandard deviationInterquartile rangeVariance

Alex Tutor

The median splits sorted data in half. Quartiles split it into quarters: Q1 has a quarter of the values below it, Q2 is the median, and Q3 has three quarters below it. A simple way to find them by hand: Q1 is the median of the lower half of the data, Q3 the median of the upper half.

Courier B, sorted: 14, 14, 18, 18, 24 | 24, 30, 30, 30, 38. The lower half is 14, 14, 18, 18, 24, so Q1 = 18. The upper half is 24, 30, 30, 30, 38, so Q3 = 30.

The interquartile range (IQR) is

Q_3 - Q_1 = 30 - 18 = 12

hours: the width of the middle 50% of deliveries. Unlike the range and the SD, the IQR ignores the extremes completely. That lost 90-hour parcel wouldn't change it at all. That is why IQR is the spread partner of the median, and SD the partner of the mean.

Quartiles are special cases of percentiles: the 90th percentile is the value below which 90% of the data falls. Product teams use them all the time: "p90 delivery time is 30 hours", "p95 page load is 2.8 seconds". They describe the experience of your unluckiest customers, which the average hides.

Alex Tutor

\text{IQR} = Q_3 - Q_1

Interquartile range: the spread of the middle 50% of the data.

Alex Tutor

Question 6

Daily ad spend: mean $2,000, SD $200. Daily sign-ups from ads: mean 50, SD 15. Which is more variable relative to its size?

Choose one answerAd spend, because its SD is largerSign-ups: CV 30% vs 10%They are equally variableThey can't be compared

Alex Tutor

Is an SD of 8 big? It depends on the scale. 8 hours is a lot for a 24-hour delivery, but trivial for a 30-day subscription cycle. To compare spread across metrics with different units or sizes, divide the SD by the mean. That's the coefficient of variation (CV), usually shown as a percentage.

Courier B:

8 / 24 \approx 33\%

. Now a marketing example: CatChow's daily ad spend averages $1,000 with an SD of $150 (CV 15%), while daily orders from ads average 30 with an SD of 9 (CV 30%). Spend is steady, yet results swing twice as much, relative to their size. That's a hint that something other than budget (day of week, creatives, stock-outs) drives the orders.

CV only makes sense for values measured from a true zero, like hours, money or counts. Don't use it for things like temperature or NPS, which can be negative.

Alex Tutor

Question 7Short answer · AI-checked task

A PM reports: "Average checkout time is 40 seconds (SD 35 seconds)." What does the large SD tell you, and which extra number would you ask for? Answer in 2–4 sentences.

Write your answer and get a score with feedback from our AI reviewer.

Log in to get AI feedback

Alex Tutor

\text{CV} = \frac{s}{\bar{x}} \times 100\%

Coefficient of variation: SD as a share of the mean.

Alex Tutor

Measure	Uses	Robust to extremes?	Good for
Range	2 values	No	A first glance, "best and worst case"
Variance	All values	No	Maths inside other methods (you'll meet it in tests and ANOVA)
Standard deviation	All values	No	Reporting spread with the mean: "24 ± 8 hours"
IQR / percentiles	Order of values	Yes	Skewed data, SLAs, reporting with the median
CV	SD and mean	No	Comparing spread across metrics on different scales

Back to the couriers. Courier A: range 5 hours, and every delivery within the promise. Courier B: range 24 hours, SD 8 hours, IQR 12 hours, and 1 of 10 deliveries late. Same average, very different customer experience. The case task at the end asks you to make the call.

Alex Tutor

Question 8Case · AI-checked task

Which courier should CatChow keep, and why? Back your answer with numbers from the dataset.

CatChow promises customers delivery within 36 hours. The operations lead wants to keep just one courier. Her summary: "Both average 24 hours, so let's keep courier B: it's 10% cheaper."

You have the last 10 deliveries for each courier (dataset CatChow deliveries by courier). Write a short recommendation: what the average hides, which spread measures you'd look at, which courier you'd keep, and what else you'd want to know before signing.

Dataset for this task: CatChow deliveries by courier

Write your answer and get a score with feedback from our AI reviewer.

Log in to get AI feedback

Alex Tutor

That’s the lesson. You answered every task — nicely done.

Next lesson: Outliers and skewed data