Year 11 General Univariate Data Analysis

Characteristics Of Distributions Of Numerical Data: Shape, Location And Spread

Q: What are the three things to describe about a distribution?

Shape (the overall pattern — symmetric, skewed, or bimodal), location (where the centre sits — mean, median, mode), and spread (how widely the values vary — range, IQR, standard deviation).

Q: What's the difference between positively and negatively skewed?

Positively skewed (right-skewed) has a long tail on the right; the mean is greater than the median because large values pull it up. Negatively skewed (left-skewed) has a long tail on the left; the mean is less than the median.

Q: How do I calculate the median for an even number of values?

Sort the values from smallest to largest. The median is the average of the two middle values. For 8 values, the median is the average of the 4th and 5th values.

Q: What is the interquartile range and why is it useful?

The interquartile range (IQR) is Q3 minus Q1 — the spread of the middle 50 percent of the data. It is resistant to outliers, unlike the range and standard deviation, which makes it a good measure of spread for skewed data.

Q: When should I use mean vs median?

Use the mean (with standard deviation) for roughly symmetric data with no outliers. Use the median (with IQR) for skewed data or data with outliers, because the median is barely affected by extreme values.

Q: What is the five-number summary?

The five-number summary is: minimum, Q1, median, Q3, maximum. It is the foundation for drawing a boxplot and gives a compact picture of the centre, spread, and tails of the distribution.

20 practice questions 2 video lessons Theory + worked examples

Theory

Describe a distribution by shape (symmetric, positively skewed, negatively skewed), location (mean, median, mode), and spread (range, IQR, standard deviation). The mean is pulled by outliers; the median is not. Use mean + standard deviation for symmetric data, median + IQR for skewed data or data with outliers.

A numerical distribution is summarised by three features:

Shape — the overall pattern. Is it symmetric, skewed to one side, or has multiple peaks?
Location (centre) — where the typical value sits.
Spread — how widely the values vary.

Common shapes:

Symmetric — left and right halves mirror each other. Mean $\approx$ median.
Positively skewed (right-skewed) — long tail on the right. Mean $>$ median.
Negatively skewed (left-skewed) — long tail on the left. Mean $<$ median.
Bimodal — two distinct peaks (often two mixed groups).

Common location measures:

Mean $\bar{x} = \frac{\sum x}{n}$ . Sensitive to outliers.
Median — middle value when sorted. Position $= \frac{n + 1}{2}$ . For even $n$ , average the two middle values. Resistant to outliers.
Mode — most frequent value.

Common spread measures:

Range = max $-$ min.
Quartiles $Q_{1}, Q_{2}, Q_{3}$ split sorted data into four equal parts; $Q_{2}$ is the median.
Interquartile range: $IQR = Q_{3} - Q_{1}$ . Spread of the middle 50%. Resistant to outliers.
Standard deviation $s$ — typical distance from the mean. Larger $s$ = more spread.
Five-number summary: min, $Q_{1}$ , median, $Q_{3}$ , max.

The first diagram shows the three main shapes of a distribution. The second demonstrates how an outlier pulls the mean but barely moves the median.

Symmetric shapes use mean + standard deviation; skewed shapes use median + IQR.

Adding

50

{5, 6, 7, 8, 9}

shifts the mean to

\approx 14

; the median moves only from

7

7.5

The key formulas are for the mean, the median position, and the IQR.

Mean

$\bar{x} = \frac{\sum x}{n}$

\bar{x} = \frac{\sum x}{n}

Median position

For sorted data of size $n$ , the median is at position $\frac{n + 1}{2}$ . For odd $n$ , this gives a single position. For even $n$ , average the two middle values $\frac{a + b}{2}$ .

position = \frac{n + 1}{2}

Spread

$range = max - min, IQR = Q_{3} - Q_{1}$

IQR = Q_{3} - Q_{1}

Choosing the right summary

Shape	Centre	Spread
Symmetric, no outliers	Mean $\bar{x}$	Standard deviation $s$
Skewed or with outliers	Median	IQR

The five-number summary: min, Q1, median, Q3, max. These five values capture centre, spread, and tail behaviour and are the basis for a boxplot.

Calculating the median

Sort the data from smallest to largest.
Find position $\frac{n + 1}{2}$ .
For odd $n$ : median is the value at that position. For even $n$ : average the two middle values.

Calculating quartiles and IQR

Sort the data and find the median ( $Q_{2}$ ).
Lower half (everything below the median): $Q_{1}$ = median of this half.
Upper half (everything above the median): $Q_{3}$ = median of this half.
$IQR = Q_{3} - Q_{1}$ .

Choosing centre and spread

Look at the shape first — is it symmetric, skewed, or contain outliers?
If symmetric and no outliers: use mean + standard deviation.
If skewed or has outliers: use median + IQR. These are resistant to extreme values.

EXAMPLE 1 — MEAN

Hours of homework for

10

students:

8, 12, 15, 10, 9, 14, 11, 13, 16, 12

. Find the mean.

SOLUTION

Add all values, then divide by $n = 10$ .

$\sum x$	$=$	$120$
$\bar{x}$	$=$	$\frac{120}{10}$
	$=$	$12 hours$

Answer: the mean is $12$ hours of homework.

\bar{x} = 12

EXAMPLE 2 — MEDIAN (EVEN n)

Cricket scores (sorted):

12, 18, 25, 33, 41, 47, 52, 60

. Find the median.

SOLUTION

$n = 8$ , so the median is the average of the $4$ th and $5$ th values.

Median	$=$	$\frac{33 + 41}{2}$
	$=$	$37$

Answer: the median score is $37$ .

median = 37

EXAMPLE 3 — IQR AND FIVE-NUMBER SUMMARY

Hours of part-time work (sorted):

12, 15, 18, 20, 22, 25, 28, 30, 32, 35, 38

. Find

Q_{1}

Q_{3}

, the IQR, and the five-number summary.

SOLUTION

$n = 11$ (odd), so the median is the $6$ th value, which is $25$ . Lower half (values 1–5): $12, 15, 18, 20, 22$ . Upper half (values 7–11): $28, 30, 32, 35, 38$ .

$Q_{1}$	$=$	median of $12, 15, 18, 20, 22$ $= 18$
$Q_{3}$	$=$	median of $28, 30, 32, 35, 38$ $= 32$
IQR	$=$	$32 - 18 = 14$

Five-number summary: $12, 18, 25, 32, 38$ .

Answer: $Q_{1} = 18$ , $Q_{3} = 32$ , $IQR = 14$ ; five-number summary is $12, 18, 25, 32, 38$ .

IQR = 14

EXAMPLE 4 — EFFECT OF AN OUTLIER

The dataset

5, 6, 7, 8, 9

has mean

7

and median

7

. An extra value of

50

is added. Find the new mean and new median, and compare.

SOLUTION

New dataset: $5, 6, 7, 8, 9, 50$ ; $n = 6$ .

New mean	$=$	$\frac{5 + 6 + 7 + 8 + 9 + 50}{6}$
	$=$	$\frac{85}{6} \approx 14.17$
New median	$=$	$\frac{7 + 8}{2} = 7.5$

The mean jumped from $7$ to about $14.17$ — a huge shift. The median moved only from $7$ to $7.5$ .

Answer: the outlier pulled the mean dramatically but barely moved the median, illustrating why the median is preferred for skewed or outlier-prone data.

new mean \approx 14.17, new median = 7.5

Common pitfalls

The mean is dragged by outliers. Adding one extreme value can shift the mean a lot but barely move the median. If a dataset has outliers, use the median + IQR rather than the mean + standard deviation.

Median rule for even $n$ . When the dataset has an even number of values, the median is the average of the two middle values, not just the lower or upper one. Write

\frac{a + b}{2}

clearly in your working.

Sort before finding the median or quartiles. If the data isn't sorted, the "middle value" is meaningless. Always sort from smallest to largest first.

Quartiles split the data, not the values.

Q_{1}

is the median of the lower half of the data, not the smallest quarter of the range of values. They can give very different answers.

Five-number summary order. Always list: minimum,

Q_{1}

, median,

Q_{3}

, maximum — in that order. Don't put the IQR or the range in this list.

Frequently asked questions

What are the three things to describe about a distribution?

Shape (the overall pattern — symmetric, skewed, or bimodal), location (where the centre sits — mean, median, mode), and spread (how widely the values vary — range, IQR, standard deviation).

What's the difference between positively and negatively skewed?

Positively skewed (right-skewed) has a long tail on the right; the mean is greater than the median because large values pull it up. Negatively skewed (left-skewed) has a long tail on the left; the mean is less than the median.

How do I calculate the median for an even number of values?

Sort the values from smallest to largest. The median is the average of the two middle values. For 8 values, the median is the average of the 4th and 5th values.

What is the interquartile range and why is it useful?

The interquartile range (IQR) is Q3 minus Q1 — the spread of the middle 50 percent of the data. It is resistant to outliers, unlike the range and standard deviation, which makes it a good measure of spread for skewed data.

When should I use mean vs median?

Use the mean (with standard deviation) for roughly symmetric data with no outliers. Use the median (with IQR) for skewed data or data with outliers, because the median is barely affected by extreme values.

What is the five-number summary?

The five-number summary is: minimum, Q1, median, Q3, maximum. It is the foundation for drawing a boxplot and gives a compact picture of the centre, spread, and tails of the distribution.

Video Lessons

Practice Questions

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Characteristics Of Distributions Of Numerical Data: Shape, Location And Spread

📖 Theory

Mean

Median position

Spread

Choosing the right summary

Calculating the median

Calculating quartiles and IQR

Choosing centre and spread

Common pitfalls

Frequently asked questions

🎬 Video Lessons

✏️ Practice Questions

Theory

Video Lessons

Practice Questions