Descriptive statistics: measures of center and spread

Q: When should I use median instead of mean?

Use median when your data is heavily skewed or has extreme outliers, because the median is not affected by extreme values, while the mean is pulled towards the outlier. Median gives a better representation of the typical value in these cases.

Q: Why do we divide by n-1 for sample standard deviation?

Dividing by n-1 corrects for bias in the estimate of the population variance when working with a sample. This Bessel correction ensures your sample estimate is closer to the true population value.

IB Mathematics: Applications and Interpretation HL· 30 min read

1. Measures of Central Tendency (Center)★☆☆☆☆⏱ 8 min

📘 Definition

Measure of center

A single value that describes the central position of a set of numerical data, also called a measure of central tendency

Example:

The average height of students in a class is a measure of center

Three common measures of center are used for descriptive statistics, each suited to different contexts:

Mean: The arithmetic average of all data values, calculated as the sum of all values divided by the number of values.
Median: The middle value of an ordered data set, with half the values above and half below.
Mode: The most frequently occurring value in a data set, can be used for categorical as well as numerical data.

📐 Worked Example

A small café records the number of customers per hour over 6 hours: [12, 8, 15, 10, 8, 14]. Calculate the mean, median and mode.

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Step 1: Calculate the mean by summing all values and dividing by the number of observations:
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$\frac{12 + 8 + 15 + 10 + 8 + 14}{6} = \frac{67}{6} \approx 11.17$
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Step 2: Order the data set from smallest to largest:
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$[8, 8, 10, 12, 14, 15]$
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Step 3: For an even number of observations, the median is the average of the two middle values:
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$\text{Median} = \frac{10 + 12}{2} = 11$
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Step 4: The mode is the most frequently occurring value:
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Mode = 8 customers

2. Measures of Dispersion (Spread)★★☆☆☆⏱ 10 min

📘 Definition

Measure of spread

A value that describes how far apart data points are from each other and from the center of the distribution

Example:

The standard deviation of test scores tells you how much scores vary around the average

Common measures of spread quantify the variation in a data set:

Range: Difference between the maximum and minimum value, simple but very sensitive to outliers.
Interquartile Range (IQR): Difference between the third quartile (Q3, 75th percentile) and first quartile (Q1, 25th percentile), not sensitive to outliers.
Variance: Average of the squared deviations from the mean.
Standard deviation: Square root of variance, measured in the same units as the original data.

📐 Worked Example

For the ordered customer data set [8, 8, 10, 12, 14, 15], calculate the IQR and sample standard deviation.

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Step 1: Find quartiles: Q1 is the median of the lower half, Q3 is the median of the upper half:
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$Q1 = \frac{8 + 8}{2} = 8, \quad Q3 = \frac{14 + 15}{2} = 14.5$
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Step 2: Calculate IQR = Q3 - Q1:
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$IQR = 14.5 - 8 = 6.5$
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Step 3: Use the sample standard deviation formula, with sample mean $\overset{x}{ˉ} \approx 11.17$ :
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$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$
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Step 4: Calculate the sum of squared deviations, divide by $n - 1 = 5$ , then take the square root:
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$s \approx 2.99$

Exam tip:

Always check if the question asks for population or sample standard deviation — marks are deducted for the wrong output from your GDC.

3. Selecting Appropriate Measures★★☆☆☆⏱ 7 min

The choice of which measure to use depends entirely on the shape of your distribution and the presence of outliers.

📐 Worked Example

A company has 5 employees with annual salaries (in thousands of dollars): [45, 48, 52, 55, 250]. Which measure of center is most appropriate for describing the typical employee salary?

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Step 1: Identify that 250 is an extreme outlier, meaning the distribution is heavily right-skewed.
2
Step 2: Calculate both mean and median for comparison:
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$\text{Mean} = 90, \quad \text{Median} = 52$
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Step 3: The mean is pulled up by the outlier to $90,000, which is higher than 4 out of 5 salaries. The median of $52,000 is much closer to the typical salary, so it is the appropriate measure.

4. Effect of Linear Transformations★★★☆☆⏱ 5 min

When you apply a linear transformation of the form $y = k x + c$ to all data points (for example, converting units from cm to inches), measures of center and spread change in predictable ways:

Adding a constant $c$ to all values: adds $c$ to all measures of center, does not change any measures of spread.
Multiplying all values by a positive constant $k$ : multiplies all measures of center by $k$ , multiplies all measures of spread by $k$ , and multiplies variance by $k^{2}$ .

📐 Worked Example

The mean of a data set of lengths is 25 cm, with standard deviation 3 cm. Find the new mean and standard deviation after converting to inches (1 inch = 2.54 cm).

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Step 1: The transformation is $x_{inch} = \frac{x _{cm}}{2.54}$ , so $k = \frac{1}{2.54} \approx 0.3937$ .
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Step 2: The new mean equals the old mean multiplied by $k$ :
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$\text{New Mean} = 25 \times 0.3937 \approx 9.84 \text{ inches}$
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Step 3: The new standard deviation equals the old standard deviation multiplied by $k$ :
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$\text{New SD} = 3 \times 0.3937 \approx 1.18 \text{ inches}$

5. Histograms and Cumulative Frequency Graphs★★☆☆☆⏱ 12 min

For larger data sets, raw lists are replaced by frequency distributions. Two standard graphs summarize grouped data: the histogram, which reveals the shape of the distribution, and the cumulative frequency graph, which lets you read off the median, quartiles and percentiles.

📘 Definition

Histogram

A graph for continuous (grouped) data where adjacent bars represent class intervals. For equal-width classes the bar height is the frequency; for unequal-width classes the height is the frequency density (frequency divided by class width).

Example:

A histogram of 80 students' test scores in classes of width 20 shows the distribution is roughly symmetric and unimodal.

📘 Definition

Cumulative frequency graph

Plotting the cumulative frequency (running total of frequencies) against the upper boundary of each class and joining the points with a smooth curve produces an S-shaped ogive.

Example:

If 5 students scored at most 20 and 15 more scored at most 40, the cumulative frequency at a score of 40 is 20.

To estimate positional measures from a cumulative frequency graph with $n$ data values, read the $x$ -value at these cumulative frequencies:

Median: read across from cumulative frequency $\frac{n}{2}$ .
Lower quartile Q1: read across from $\frac{n}{4}$ .
Upper quartile Q3: read across from $\frac{3 n}{4}$ .
$k$ -th percentile: read across from $\frac{k}{100} \times n$ .
Then IQR $= Q 3 - Q 1$ .

Score interval	Frequency	Upper boundary	Cumulative frequency
$0 < x \leq 20$	5	20	5
$20 < x \leq 40$	15	40	20
$40 < x \leq 60$	30	60	50
$60 < x \leq 80$	20	80	70
$80 < x \leq 100$	10	100	80

📐 Worked Example

Using the cumulative frequency table above for the test scores of 80 students, estimate the median, the quartiles and the interquartile range.

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Step 1: With $n = 80$ , the median is at cumulative frequency $\frac{n}{2} = 40$ . This falls in class $40 < x \leq 60$ , between cumulative frequency 20 (at 40) and 50 (at 60). Interpolate:
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$\text{Median} \approx 40 + \frac{40 - 20}{50 - 20} \times 20 \approx 53.3$
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Step 2: Q1 is at $\frac{n}{4} = 20$ . Cumulative frequency 20 occurs exactly at a score of 40, so $Q 1 \approx 40$ .
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Step 3: Q3 is at $\frac{3 n}{4} = 60$ , which falls in class $60 < x \leq 80$ , between cumulative frequency 50 (at 60) and 70 (at 80). Interpolate:
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$Q3 \approx 60 + \frac{60 - 50}{70 - 50} \times 20 = 70$
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Step 4: Calculate the interquartile range:
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$IQR = Q3 - Q1 = 70 - 40 = 30$

Exam tip:

On a cumulative frequency graph, read the median off at cumulative frequency $\frac{n}{2}$ , Q1 at $\frac{n}{4}$ and Q3 at $\frac{3 n}{4}$ — use $n$ itself, not $n + 1$ , for grouped data in AI.

6. Common Pitfalls

Wrong move:

Confusing population and sample standard deviation on your GDC output

Why:

Most graphic display calculators output both values with different symbols, and exams require the correct value for the context

Correct move:

Check the question: if working with a sample (most cases in AI HL), use $s$ , if working with a full population, use $σ$

Wrong move:

Using mean and standard deviation for heavily skewed data with outliers

Why:

Outliers pull the mean away from the true center of the data, leading to a misleading description of the typical value

Correct move:

Check for skewness and outliers first: if present, use median and IQR instead

Wrong move:

Calculating the median from an unordered data set

Why:

The median is defined as the middle value of sorted data, so taking the middle value of unsorted data will almost always be wrong

Correct move:

Always sort the data set from lowest to highest before calculating the median or quartiles

Wrong move:

Confusing range and interquartile range

Why:

Many students incorrectly calculate IQR as maximum minus minimum, which is actually range

Correct move:

IQR is Q3 minus Q1, it measures the spread of the middle 50% of the data

Wrong move:

Thinking adding a constant to all data changes the spread

Why:

Students often assume adding a constant changes both center and spread

Correct move:

Adding a constant shifts all points equally, so the distance between points (spread) stays exactly the same

7. Quick Reference Cheatsheet

Measure Type	Name	Best Used For	Transformation $x \to k x + c$
Center	Mean	Symmetric data, no outliers	$\to k μ + c$
Center	Median	Skewed data, with outliers	$\to k M + c$
Center	Mode	Categorical/bimodal data	$\to k m + c$
Spread	Range	Rough quick estimate	$\to k \times Range$
Spread	IQR	Skewed data, with outliers	$\to k \times I QR$
Spread	SD	Symmetric data, no outliers	$\to k \times σ$
Spread	Variance	Statistical calculations	$\to k^{2} \times σ^{2}$

8. Frequently Asked

When should I use median instead of mean?

Use median when your data is heavily skewed or has extreme outliers, because the median is not affected by extreme values, while the mean is pulled towards the outlier. Median gives a better representation of the typical value in these cases.

Why do we divide by n-1 for sample standard deviation?

Dividing by n-1 corrects for bias in the estimate of the population variance when working with a sample. This Bessel correction ensures your sample estimate is closer to the true population value.

What's Next

Understanding measures of center and spread is the foundation for all further statistics topics in IB AI HL. You will next use these concepts to analyze bivariate data, where you will calculate the center and spread of variables for correlation and regression. Later, you will use these measures to describe probability distributions, and for statistical inference, where the center and spread of sampling distributions are critical for constructing confidence intervals and running hypothesis tests.