Basic statistics · Lesson 4 of 10

Spread: range, variance and standard deviation

An average tells you where the data are centred and nothing about how far they stray. Two districts can share a mean income and still be very different places to live. This lesson builds the measures of spread from the simplest to the standard deviation, shows what shifting and rescaling do to each, and ends with how to combine groups.

← Basic statistics moduleLesson 4 of 10 · about 60 minutes

A question to hold on to

Sample A is −6, −6, −1, 0, 1, 6, 6. Sample B is 521, 524, 526, 527, 528, 530, 533. Which has the larger range, and which the larger variance?

Hyderabad 2021Solved at the end ↓

New here?What is Ashanagar?The invented district behind this page's examples: 60 households, with incomes you can inspect.

Ashanagar is an invented district. The statistics lessons use it for every worked example, so you meet the same households again as the ideas build: first as a list of numbers, then as a spread, then as a population to sample from.

It has 60 households, each with a monthly income in rupees. Unless a page says otherwise, those 60 are the whole population, and a “sample” means a few of them drawn at random. Where a page needs two variables, it uses a smaller set of 12 households with the head’s years of schooling beside income.

The figures are illustrative, made up once and fixed. They are not official statistics.

Households: 60
Mean income: ₹28,783 a month
Median income: ₹25,000 a month
Standard deviation: ₹13,366
Lowest and highest: ₹10,500 and ₹72,000

Meet the full data set →

The idea

Same mean, different lives

Here are five households in each of two districts. Both have a mean income of ₹25,000 a month. Would you rather live in A or in B?

District A: five households, income in ₹ thousand a month

District B: five households, same mean

District A: mean, standard deviation25, 3.22

District B: mean, standard deviation25, 11.4

Both districts have a mean of ₹25,000. In A nobody is far from it. In B nobody is within ten thousand rupees of it except one household. The mean alone cannot tell them apart; a measure of spread can.

The mean cannot tell them apart; a measure of dispersion, or spread, can. That is not a technicality. Spread is how economists talk about inequality (how far incomes are from each other), risk (how far a return can stray from its average) and instability (how far prices swing around their usual level).

Three quick measures

The simple measures: range, quartile deviation, mean deviation

Range = largest − smallest. CUET 2020: rice prices of 350, 280, 340, 290, 320, 310 and 300 have a range of 70. It is instant, but it uses only two values, so one freak observation changes it completely. It is used where the extremes are exactly what matter: a day’s temperature band, a share price range, quality-control limits.
Interquartile range = Q₃ − Q₁, the spread of the middle half of the data. Half of the observations lie between the quartiles (JNU 2016). It ignores the top and bottom quarter, so extreme values cannot move it, and that is why CUET 2018 and Hyderabad 2021 give it as the answer to “which is unaffected by outliers?”
Quartile deviation, or semi-interquartile range, is half of that: (Q₃ − Q₁) ÷ 2. If Q₁ = 142 and the quartile deviation is 18, then Q₃ = 178, and for a symmetrical distribution the median sits halfway at 160 (Hyderabad 2021). Because it needs only the quartiles, it works when the top class is open-ended, as Hyderabad 2014 asks.
Mean deviation = the average of the absolute deviations, $\frac{1}{n} \sum ∣ x_{i} - x ˉ ∣$ . It uses every value. About the median it is smaller still, as Lesson 2 showed.

In Ashanagar the range of incomes is ₹61,500 (₹10,500 to ₹72,000), the quartiles are ₹20,000 and ₹38,375, so the interquartile range is ₹18,375 and the quartile deviation ₹9,187.50. The range is driven by the one household earning ₹72,000. The interquartile range is not.

Absolute and relative measures

All four above are absolute measures: they are in the units of the data, rupees here, so they cannot compare two different variables. A relative measure divides by a measure of size and has no units. The coefficient of quartile deviation, (Q₃ − Q₁) ÷ (Q₃ + Q₁), is one: with Q₃ = 40 and Q₁ = 20 it is 20 ÷ 60 = 0.33 (CUET 2021). The most important is the coefficient of variation, below. Hyderabad 2018 and JAM 2026 both ask which of several measures is relative.

Interactive

Variance and standard deviation

The mean deviation is natural, but the absolute value in it is awkward to work with algebraically. Statisticians square the deviations instead, and the two main measures of spread both come from that. Each comes in two versions, one for a population (you have every value) and one for a sample (you have some, and want to estimate the population’s spread). The letters are different, so learn them as pairs.

Variance

The average squared distance from the mean. It is in squared units (₹², kg²), which is why we also use its square root.

Population variance σ²: $σ^{2} = \frac{\sum _{i = 1}^{n} ( x _{i} - μ ) ^{2}}{n}$ Every value is in the data, so divide by n. Here μ is the population mean.
Sample variance s²: $s^{2} = \frac{\sum _{i = 1}^{n} ( x _{i} - x ˉ ) ^{2}}{n - 1}$ The data are a sample, so divide by n − 1. Here x̄ is the sample mean.

Standard deviation

The square root of the variance: a typical distance from the mean, in the same units as the data.

Population standard deviation σ: $σ = σ^{2} = \frac{\sum _{i = 1}^{n} ( x _{i} - μ ) ^{2}}{n}$ The root of the population variance. Divide by n inside the root.
Sample standard deviation s: $s = s^{2} = \frac{\sum _{i = 1}^{n} ( x _{i} - x ˉ ) ^{2}}{n - 1}$ The root of the sample variance. Divide by n − 1 inside the root.

Both start the same way: subtract the mean from each value, square the result so that every term is positive and a value twice as far out counts four times as much, and add the squares up. Only the divisor differs, n for a population and n − 1 for a sample. The next section explains why.

The picture makes it concrete. Below, six markets quote an onion price. Each price’s distance from the mean is drawn as a square: its side is the deviation, so its area is the squared deviation. The variance is the average of the areas, and the dashed square at the end has exactly that area. Its side is the standard deviation. Drag a price out and watch the squares grow. The buttons under the picture switch between the two versions: divide by n when the six markets are the whole population, divide by n − 1 when they are only a sample of markets.

Mean25

Squares added up310

Population variance σ² (÷ 6)51.67

Population standard deviation σ7.19

Blue squares are values above the mean, red below. A value twice as far from the mean makes a square four times as big, which is why one distant value moves the variance so much more than several close ones.

What the standard deviation says

It is a typical distance from the mean, in the same units as the data: ₹ for incomes, kg for weights. A standard deviation of ₹13,000 says that households in Ashanagar are typically about ₹13,000 away from the average income, in one direction or the other.

A shortcut for calculation

Squaring all those deviations is slow. Expand the square and the variance becomes the mean of the squares minus the square of the mean:

Shortcut formula

σ^{2} = \frac{\sum x _{i}^{2}}{n} - x ˉ^{2}

JNU 2020 · standard

The numbers of defective mobiles in four shipments are 2, 4, 6 and 8. Find the mean and standard deviation of this population.

Mean.
(2 + 4 + 6 + 8) ÷ 4 = 5.
Deviations and squares.
−3, −1, 1, 3, with squares 9, 1, 1, 9. They add to 20.
Variance.
20 ÷ 4 = 5. By the shortcut: (4 + 16 + 36 + 64) ÷ 4 − 25 = 30 − 25 = 5. The same.
Standard deviation.
√5 = 2.24.

Answer

Mean 5 and standard deviation √5. The question says population, so the divisor is 4. Dividing by 4 − 1 = 3 would give a variance of 6.67, which is not among the options.

DSE 2009 · Delhi School level

Twenty passengers are found without tickets. The sum of squares and the population standard deviation of the money in their pockets are ₹2,000 and ₹6. The total fine equals the total money found, shared equally. What is each fine?

Use the shortcut.
σ² = Σx² ÷ n − x̄², so 36 = 2,000 ÷ 20 − x̄² = 100 − x̄².
Solve for the mean.
x̄² = 64, so the mean amount is ₹8.
The fine.
Total money = 20 × 8 = ₹160, shared among 20 people: ₹8 each.

Answer

₹8. The shortcut formula is the only way in: the question gives a sum of squares and a standard deviation, never the mean.

A few facts that examiners lean on:

The standard deviation of two values is half their difference (JNU 2021): for x₁ and x₂ the mean is halfway and each is |x₁ − x₂| ÷ 2 away.
If the variance is zero, every value equals the mean: all the elements of a population with mean 25 and variance 0 are 25 (JNU 2015).
With the mean at 4, the values 2, 3, 4, 5, 6 have squared deviations adding to 4 + 1 + 0 + 1 + 4 = 10 (CUET 2020).
A larger standard deviation does not mean a larger range, or more skewness. It only says the values are spread more on average (JNU 2015).

Interactive

Why a sample divides by n − 1

If the data are the whole population, divide by n. If they are a sample used to estimate the population’s spread, divide by n − 1:

Sample variance

s^{2} = \frac{1}{n - 1} i = 1 \sum n (x_{i} - x ˉ)^{2}

CUET 2025 asks for the standard deviation of the sample 8, 7, 9. The mean is 8, the squared deviations are 0, 1, 1, and 2 ÷ (3 − 1) = 1, so s = 1.

Why n − 1? The deviations are measured from the sample’s own mean, which is the point that makes them as small as possible (Lesson 2). A sample’s values always sit a bit closer to their own mean than to the true population mean, so dividing by n gives an estimate that is too small, on average. Dividing by n − 1 corrects it exactly. A second way to see it: the deviations add to zero, so once n − 1 of them are known the last is fixed. Only n − 1 are free, and that count is called the degrees of freedom.

Remember the pairing: n for a population, n − 1 for a sample. The lab uses the sixty Ashanagar incomes as a population whose true variance σ² is known. Each draw takes a sample of n households at random and works out its variance both ways, once with the population formula (÷ n) and once with the sample formula (÷ n − 1). The bars show the average over 3,000 draws. Only the sample formula lands on σ².

Households in each sample, n5Each draw takes 5 households at random, with replacement

Draw set 1

Population variance σ² of all 60 households (the truth)100%

Average of the ÷ n estimates (the population formula, wrongly used on a sample)80.6%

Average of the ÷ (n − 1) estimates (the sample formula, s²)100.8%

With samples of 5, dividing by n gives estimates that average 80.6% of the truth. The shortfall is about (n − 1)/n = 80%, because a sample’s values sit closer to their own mean than to the population’s. Dividing by n − 1 corrects it, and the average comes to 100.8%. Draw again to see the figures wobble a little, but not the gap.

Derive it

Where n − 1 comes from

We want to show

The sum of squared deviations from the sample mean averages (n − 1)σ², not nσ², so s² = Σ(x − x̄)² ÷ (n − 1) averages σ².

Start from the ideal.
If we knew the true mean μ we would measure each deviation from it. Each (x_i − μ)² has an average of σ², which is what σ² means, so the sum of n of them has an average of nσ². We do not know μ, so we measure from the sample mean x̄ instead. The question is what that costs.
Slip x̄ into the middle of each deviation.
Write x_i − μ = (x_i − x̄) + (x̄ − μ), square it and add over the sample:
$\sum (x_{i} - μ)^{2} = \sum (x_{i} - x ˉ)^{2} + 2 (x ˉ - μ) \sum (x_{i} - x ˉ) + n (x ˉ - μ)^{2}$
The middle term has Σ(x_i − x̄), and deviations from a sample’s own mean always add to zero, so it disappears:
$\sum (x_{i} - μ)^{2} = \sum (x_{i} - x ˉ)^{2} + n (x ˉ - μ)^{2}$
Measuring from x̄ always gives less.
Rearranged, Σ(x_i − x̄)² = Σ(x_i − μ)² − n(x̄ − μ)². The piece taken away is a square, so it is never negative: a sample sits closer to its own mean than to the population mean. Draw some samples and watch the blue part fall short of the whole by exactly the red piece.
How big is the lost piece on average?
The sample mean x̄ varies from sample to sample with variance σ² ÷ n (shown in the standard error derivation of the inference module). So n(x̄ − μ)² averages n × σ² ÷ n = σ². One σ², whatever the sample size.
Subtract, then divide.
$E [\sum (x_{i} - x ˉ)^{2}] = n σ^{2} - σ^{2} = (n - 1) σ^{2}$
To get something that averages σ² we divide by n − 1 and not by n. That is the divisor on the sample variance. It is also the meaning of degrees of freedom: n deviations from x̄ add to zero, so only n − 1 of them are free, and each free one carries one σ² on average.

Step 1 of 5

Watch out for

Which divisor?

The question says “population” or gives all the data: divide by n. JNU 2020 and the standard deviation of two values being half their difference both assume it.
The question says “sample”: divide by n − 1 (CUET 2025). Older DSE papers sometimes call the n-divisor version the sample variance, so if the question names no rule, check which one an option matches.
Only the variance of the estimator is biased, not the mean. The sample mean needs no correction.

Interactive

Shifting and scaling

This is the most frequently tested idea in the whole topic. What happens to each summary if every value x is replaced by c·x + k? Shifting by k moves the whole picture sideways. Multiplying by c stretches it. The mean and median follow both. The measures of spread notice only the stretch:

Summary	After y = cx + k	Add k only	Multiply by c only
Mean, median	c · (old) + k	+ k	× c
Range, quartile deviation, mean deviation, SD	\|c\| · (old)	unchanged	× \|c\|
Variance	c² · (old)	unchanged	× c²
Coefficient of variation	\|c\|·SD ÷ \|c·mean + k\|	changes	unchanged

Add k to every value+10

Multiply every value by c× 2

Before

After: y = 2·x + 10

Summary	Before	After	Rule
Mean	25	60 changed	c · mean + k
Median	23	56 changed	c · median + k
Range	22	44 changed	\|c\| · range
Mean deviation	5.33	10.67 changed	\|c\| · mean deviation
Standard deviation	7.19	14.38 changed	\|c\| · SD
Variance	51.67	206.67 changed	c² · variance
Coefficient of variation	28.75%	23.96% changed	\|c\| · SD ÷ \|c · mean + k\|

Set k to 0 and only the multiplier is left: everything scales except the coefficient of variation. Set c to 1 and only k is left: the mean and median shift and the spread does not move at all, so the coefficient of variation falls when you add and rises when you subtract.

The questions are all one idea in different clothes:

Add 5 to every value of a data set with mean 10 and SD 1: mean 15, SD 1 (JNU 2011). Add 1 kg to every child’s weight: mean 31, SD 3 (Hyderabad 2011).
The variance of 15 observations is 4; add 9 to each: still 4 (JNU 2021).
If Y = 20 + 10X then Var(Y) = 100·Var(X): the 20 disappears and the 10 is squared (CUET 2024).
Multiplying by a positive number raises the variance 6.25 times: the multiplier is 2.5, which is none of the options (JNU 2014).
The numbers 1, 5, 7, 9 and 1231, 1235, 1237, 1239 have the same standard deviation, since the second is the first plus 1230. So the statement that the first set has a smaller one is the false one (JNU 2011). Likewise sample A (13, 13, 21, 29, 29) has a larger variance than B (113, 117, 121, 125, 129) even though B’s values are far larger: the level is irrelevant, only the spread is (Hyderabad 2011).

Pause and predict

UGC NET 2025 states: SD(aX) = a² · SD(X), because SD is independent of change of origin but not of scale. Is the first part correct?

Comparing across units

Relative spread: the coefficient of variation and z-scores

Is wheat or rice more volatile in price? The standard deviation cannot say, because the two prices are on different levels: a ₹5 swing matters more on a ₹30 good than on a ₹100 one. The coefficient of variation divides the standard deviation by the mean:

Coefficient of variation

CV = \frac{σ}{x ˉ} \times 100%

A pure number, usually quoted as a percentage. It measures spread as a share of the average.

A class with mean age 20 and variance 25 has a standard deviation of 5 and a CV of 25% (CUET 2018). Factory wages with a mean of ₹550 and a standard deviation of ₹110 give a CV of 20% (Hyderabad 2014). A CV of 22% on an average of 7 means a standard deviation of 0.22 × 7 = 1.54 (JNU 2021). The series with the smaller CV is the more consistent one. Ashanagar’s income CV is 46%.

Because it is a ratio, the CV is unchanged when every value is multiplied by the same number: change the currency from rupees to paise and it does not move (Hyderabad 2017). But it is changed by a shift. In a village of 100 households with mean income 400 and variance 25,600, the standard deviation is 160 and the CV is 40%. Raise every income by 100, and the mean becomes 500 while the standard deviation stays 160: the CV falls to 32%, which is a 20% decrease (Hyderabad 2020). Nothing about the households’ differences changed, and yet the coefficient did.

That behaviour is the reason for the warning in Lesson 1: the CV needs a true zero. Take the temperatures 20°C and 30°C: mean 25, standard deviation 5, CV 20%. The same two days in Fahrenheit are 68°F and 86°F: mean 77, standard deviation 9, CV 11.7%. One fact, two different “relative spreads”, because the zero of each scale is arbitrary. For income, price, quantity and other ratio-scale data the CV is safe.

z-scores

A z-score or standard score re-expresses a value as how many standard deviations it lies from the mean: z = (x − x̄) ÷ s. Standardised values have mean 0 and standard deviation 1 whatever the original units were (JNU 2012), which makes different scales comparable. A mark of 72 in a paper with mean 60 and SD 8 has z = 1.5. A mark of 65 in a paper with mean 50 and SD 5 has z = 3. The second, though lower, is much further above its class.

Putting groups together

Combining groups

Two groups with their own means and standard deviations make one bigger group. The combined mean is the weighted mean from Lesson 2. The combined variance has two parts: the spread within each group, and the spread between the groups, caused by their means being different. With group sizes n_i, means x̄_i and standard deviations σ_i, and d_i = x̄_i − the combined mean:

Combined variance

σ^{2} = \frac{\sum n _{i} ( σ _{i}^{2} + d _{i}^{2} )}{\sum n _{i}}

The d² terms are the between-group part. If the group means are equal, they vanish and the combined variance is just the weighted average of the group variances.

JNU 2016 · standard

A group of 250 items has two subgroups of 100 and 150. The first has mean 15 and SD 3. The whole group has mean 15.6 and SD √13.44. Find the SD of the second group.

Second group's mean.
Total = 250 × 15.6 = 3,900. The first group contributes 100 × 15 = 1,500, so the second has 2,400 ÷ 150 = 16.
The d terms.
d₁ = 15 − 15.6 = −0.6, so d₁² = 0.36. d₂ = 16 − 15.6 = 0.4, so d₂² = 0.16.
Write the formula.
250 × 13.44 = 100(9 + 0.36) + 150(σ₂² + 0.16), that is 3,360 = 936 + 150σ₂² + 24.
Solve.
150σ₂² = 2,400, so σ₂² = 16 and σ₂ = 4.

Answer

4. Run it forwards to check: groups (100, 15, 3) and (150, 16, 4) combine to mean 15.6 and variance (936 + 2,424) ÷ 250 = 13.44, so an SD of 3.67.

The between-group term is why the combined variance can be bigger than both group variances when the group means are far apart. Two districts each with tight incomes but very different averages make a very unequal region.

Choosing

Which measure, when?

Situation	Use
A quick look; the extremes matter	Range
Outliers, skewed data, open-ended classes	Quartile deviation or interquartile range, with the median
Standard analysis; further calculation	Standard deviation, with the mean
Comparing series with different units or levels	Coefficient of variation (ratio-scale data only)
Comparing a value across different distributions	z-score

CUET 2021 says the standard deviation is the best measure of dispersion, and it is the most widely used, because it uses every value and is the building block of everything later (correlation, regression, tests).

Two results relate the measures. For a normal distribution the standard deviation is the largest of the three and the quartile deviation the smallest: mean deviation ≈ 0.80σ and quartile deviation ≈ 0.67σ, so SD > MD > QD (Hyderabad 2016). For any distribution, Chebyshev’s inequality guarantees that at least 1 − 1/k² of the values lie within k standard deviations of the mean: at least 75% within two, and at least 89% within three. It is a floor, not a typical figure; for a normal distribution the shares are about 95% and 99.7%.

Watch out for

The mistakes examiners are fishing for

SD of a transformed variable. Scale by c and the SD changes by |c|, the variance by c². Add k and neither changes.
n or n − 1. Read whether the data are a population or a sample.
CV and a shift. The CV is unchanged by multiplying, not by adding. Adding raises the mean and lowers the CV.
Σx² against (Σx)². The shortcut formula uses the mean of the squares minus the square of the mean. Mixing them up is the commonest slip.
Same range, different variance. The range uses two values only. Two samples can match on range and differ a lot in variance.

Solving the opening question

The opening question, solved

Hyderabad 2021 · standard

Sample A: −6, −6, −1, 0, 1, 6, 6. Sample B: 521, 524, 526, 527, 528, 530, 533. Compare the ranges and the variances.

Ranges.
A: 6 − (−6) = 12. B: 533 − 521 = 12. They are equal, so any option that ranks the ranges is wrong.
Sample A's variance.
The mean is 0, so the squared deviations are the squares: 36 + 36 + 1 + 0 + 1 + 36 + 36 = 146. Dividing by 7: 20.9.
Sample B's variance.
The mean is 527. The deviations are −6, −3, −1, 0, 1, 3, 6, with squares 36, 9, 1, 0, 1, 9, 36, adding to 92. Dividing by 7: 13.1.

Answer

Equal ranges, and A has the larger variance. Dividing by n − 1 instead gives 24.3 and 15.3, the same order. B’s values cluster in the middle, while A’s sit at the ends. The range saw only the two extremes and could not tell them apart; the variance used every value.

Revise

The lesson on one screen

Range, IQR, QD: Max − min. Q₃ − Q₁. (Q₃ − Q₁) ÷ 2. Quartile measures ignore outliers and work with open-ended classes. Coefficient of QD = (Q₃ − Q₁) ÷ (Q₃ + Q₁).
Variance, SD: σ² = Σ(x − x̄)² ÷ n = Σx² ÷ n − x̄². σ = √σ². Zero only when all values are equal. SD of two values = half the difference.
Sample: Divide by n − 1 for a sample (degrees of freedom). Dividing by n underestimates the population variance by the factor (n − 1) ÷ n.
Shift and scale: y = cx + k: mean and median → c·(old) + k. Range, QD, MD, SD → |c|·(old). Variance → c²·(old). k never affects spread.
CV: σ ÷ x̄ × 100%. Unit-free; unchanged by multiplying, changed by adding; needs a true zero.
z-score: (x − x̄) ÷ s. Standardised data have mean 0 and SD 1.
Combined groups: Mean = Σ n x̄ ÷ Σn. Variance = Σ n(σ² + d²) ÷ Σn, with d = group mean − combined mean.
Relations: Normal: SD > MD > QD. Chebyshev: at least 1 − 1/k² within k SDs, for any distribution.

Practice

Check yourself

0 of 13 correct so far

Compute

Coefficient of quartile deviation

Q₁ = 12 and Q₃ = 20. Find the coefficient of quartile deviation.

Coefficient

Standard deviation

Find the population standard deviation of 2, 4, 6, 8, to two decimal places.

Sample variance

Find the sample variance of 5, 7, 9.

s²

Coefficient of variation

A series has mean 200 and variance 400. What is the coefficient of variation, in percent?

CV (%)

Shifting and scaling

A variable has mean 50 and standard deviation 5. Every value is increased by 10 and then doubled. What is the new standard deviation?

Variance of a transformation

Var(X) = 7. Find Var(3X − 4).

Variance

Shortcut formula

Five values have a sum of squares of 340 and a mean of 8. What is the variance?

σ²

z-score

A mark of 72 in a paper with mean 60 and standard deviation 8. Find the z-score.

Combined standard deviation

Group 1: 10 values, mean 20, SD 2. Group 2: 10 values, mean 30, SD 2. Find the SD of all 20 values, to two decimal places.

Quick questions

1Which of these is a relative measure of dispersion?

2Every value in a data set is increased by 7. The standard deviation

3Which measure of dispersion is best when the last class is open-ended?

4All the values of a variable are multiplied by 5. Which of these is NOT affected?

Real questions

Past papers, by level

Foundation

Definitions, one-step arithmetic and which-measure questions. Most CUET and Hyderabad questions sit here.

19 questionsPractise →

Standard

Grouped data, combined means, effects of shifting and scaling, and statement questions, up to JNU level.

23 questionsPractise →

Delhi School level

This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 10: Spread.

2 questionsLesson 10 →

All 42 past-paper questions for this lesson, easiest first: practise the full set →

These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.

← Lesson 3Geometric and harmonic means Lesson 5 →Shape: skewness, kurtosis and boxplots

Spread: range, variance and standard deviation

Same mean, different lives

The simple measures: range, quartile deviation, mean deviation

Absolute and relative measures

Variance and standard deviation

Variance

Standard deviation

A shortcut for calculation

Why a sample divides by n − 1

Which divisor?

Shifting and scaling

Relative spread: the coefficient of variation and z-scores

z-scores

Combining groups

Which measure, when?

The mistakes examiners are fishing for

The opening question, solved

The lesson on one screen

Check yourself

Compute

Quick questions

Past papers, by level

Foundation

Standard

Delhi School level

Read next

Spread: range, variance and standard deviation

Same mean, different lives

The simple measures: range, quartile deviation, mean deviation

Absolute and relative measures

Variance and standard deviation

Variance

Standard deviation

A shortcut for calculation

Why a sample divides by n − 1

Which divisor?

Shifting and scaling

Relative spread: the coefficient of variation and z-scores

z-scores

Combining groups

Which measure, when?

The mistakes examiners are fishing for

The opening question, solved

The lesson on one screen

Check yourself

Compute

Quick questions

Past papers, by level

Foundation

Standard

Delhi School level

Read next