Median and Mode

A dataset can often be summarized using a value near its centre. In the previous topic, we used the mean.

But the mean is not the only way to describe the centre of data.

Two other important measures are:

• median — the middle value
• mode — the most frequently occurring value

Together, the mean, median, and mode are called measures of central tendency or measures of centre.

Different measures answer different questions. Sometimes the mean gives the most useful description. Sometimes the median or mode tells us much more.

The key is not simply knowing how to calculate them. You need to be able to decide:

Which measure best represents the data in this situation?

What Is the Median?

The median is the middle value when a dataset is arranged in numerical order.

The word median means middle.

Consider:

3, 5, 7, 9, 11

There are five values.

The middle value is:

7

Therefore:

Median = 7

There are two values below 7 and two values above 7.

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Always Put the Data in Order First

This is the most important step when finding the median.

Consider:

12, 4, 9, 6, 15

It would be incorrect to simply select 9 because it happens to appear in the middle of the written list.

First arrange the numbers from smallest to largest:

4, 6, 9, 12, 15

Now identify the middle:

Median = 9

In this example the answer happened to remain 9, but that will not always happen.

A useful routine is:

Order → Find the Middle

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Worked Example: Odd Number of Values

Find the median:

18, 12, 21, 14, 16, 20, 15

First put the values in order:

12, 14, 15, 16, 18, 20, 21

There are seven values.

The fourth value is in the middle:

Median = 16

Three values lie below 16 and three lie above it.

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Finding the Median with an Even Number of Values

What happens if there is no single middle value?

Consider:

4, 6, 8, 10

There are four values.

The two middle values are:

6 and 8

Find the mean of those two numbers:

Median = (6 + 8) ÷ 2

Median = 14 ÷ 2

Median = 7

So when there is an even number of observations:

Order the data → Find the two middle values → Calculate their mean

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Worked Example: Even Number of Values

Find the median:

17, 8, 14, 11, 20, 9

Put the values in order:

8, 9, 11, 14, 17, 20

The middle values are:

11 and 14

Median = (11 + 14) ÷ 2

Median = 25 ÷ 2

Median = 12.5

Notice that 12.5 does not appear in the original dataset.

Like the mean, the median does not always have to be an observed value.

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Locating the Median Position

For an ordered dataset containing n values, the median position can sometimes be found using:

Median position = (n + 1) ÷ 2

For example, if there are 11 values:

Median position = (11 + 1) ÷ 2

= 12 ÷ 2

= 6

So the sixth value is the median.

For most small datasets, however, simply ordering the values and locating the middle is usually easiest.

What Does the Median Represent?

The median divides an ordered dataset approximately into two halves.

About half of the observations are at or below the median and about half are at or above it.

Suppose the ages of seven participants are:

12, 13, 14, 14, 15, 16, 17

Median = 14

The median tells us that the centre of the ordered ages is 14 years.

It does not mean everyone is 14.

It describes the location of the centre of the dataset.

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The Median Is Resistant to Extreme Values

One of the most useful features of the median is that it is less affected by unusually high or low values than the mean.

Consider:

8, 9, 10, 11, 12

Median = 10

Mean = 10

Now replace the final value with 100:

8, 9, 10, 11, 100

Median = 10

Mean = 138 ÷ 5

Mean = 27.6

The extreme value dramatically changes the mean.

But the median remains:

10

This is why the median is often useful for datasets containing outliers or extreme values.

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Median Income

Income provides a good real-world example.

Suppose five people earn:

$30,000

$32,000

$35,000

$38,000

$365,000

Mean income:

$500,000 ÷ 5 = $100,000

Median income:

$35,000

Which value better represents what a typical person in this small group earns?

In this case, the median provides useful information because the extremely high income has a large effect on the mean.

This is one reason income and house-price data are often reported using medians.

What Is the Mode?

The mode is the value that occurs most frequently in a dataset.

Consider:

2, 3, 3, 4, 5

The value 3 occurs twice.

Every other value occurs once.

Therefore:

Mode = 3

A useful way to remember this is:

Mode = Most

The mode is the value that occurs most often.

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Worked Example: Finding the Mode

Consider:

12, 14, 11, 14, 15, 14, 13

Count how often each value occurs:

11 → once
12 → once
13 → once
14 → three times
15 → once

Therefore:

Mode = 14

Unlike the median, you do not technically need to put the data in numerical order to find the mode.

However, ordering the data can make repeated values easier to identify.

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A Dataset Can Have More Than One Mode

Consider:

2, 2, 3, 4, 4, 5

The values 2 and 4 both occur twice.

No other value occurs as frequently.

Therefore, the dataset has two modes:

2 and 4

A dataset with two modes is called bimodal.

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A Dataset Can Have Several Modes

Consider:

1, 1, 2, 2, 3, 3, 4

The values:

1, 2, and 3

all occur twice.

They are all modes.

A dataset with more than two modes may be described as multimodal.

The important rule is:

Every value tied for the highest frequency is a mode.

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A Dataset Can Have No Mode

Consider:

3, 5, 7, 9, 11

Every value occurs exactly once.

No value occurs more frequently than the others.

Therefore:

There is no mode.

Do not say:

Mode = 0

Zero would only be the mode if zero actually occurred most frequently in the dataset.

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Mode with Categories

The mode has an important advantage over the mean and median.

It can be used with categorical data.

Suppose students are asked for their favourite school subject:

Science, Math, Science, English, Science, Math, Art

Science appears most frequently.

Therefore:

Mode = Science

You cannot calculate a meaningful mean of:

Science + Math + English + Art

But you can identify the mode.

This makes mode especially useful for surveys and categories.

Mode in Everyday Life

Businesses may use the mode to identify:

• most popular shoe size
• most frequently purchased product
• most common clothing size
• most popular colour
• most common order
• most frequently selected menu item

Suppose a shoe shop records these sizes sold:

7, 8, 8, 9, 8, 10, 9, 8, 11

Mode = 8

Knowing the mode may help the store decide which shoe size it needs to stock in larger quantities.

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Comparing Mean, Median, and Mode

Consider:

2, 3, 3, 4, 8

Mean

Mean = (2 + 3 + 3 + 4 + 8) ÷ 5

Mean = 20 ÷ 5

Mean = 4

Median

Ordered data:

2, 3, 3, 4, 8

Middle value:

Median = 3

Mode

Most frequent value:

Mode = 3

So:

Mean = 4

Median = 3

Mode = 3

All three describe the dataset in different ways.

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What Each Measure Tells Us

Mean

The mean uses every numerical value.

It answers:

What would each value be if the total were shared equally?

Median

The median uses the position of values.

It answers:

What is the middle of the ordered dataset?

Mode

The mode uses frequency.

It answers:

Which value or category occurs most often?

These are related ideas, but they are not interchangeable.

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Mean, Median, and Mode Can Be Equal

Consider:

2, 3, 4, 4, 4, 5, 6

Mean:

28 ÷ 7 = 4

Median:

4

Mode:

4

Therefore:

Mean = Median = Mode = 4

This often happens in datasets that are reasonably balanced around a central value.

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Mean, Median, and Mode Can Be Different

Consider:

2, 3, 3, 4, 18

Mean:

30 ÷ 5 = 6

Median:

3

Mode:

3

The extreme value 18 has pulled the mean upward.

This difference between the measures gives us useful information about the shape of the dataset.

Choosing the Best Measure of Centre

There is no rule saying that one measure is always best.

The appropriate measure depends on:

• the type of data
• the distribution of the values
• whether extreme values are present
• what question you are trying to answer

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When the Mean Is Useful

The mean is often useful when:

• the data are numerical
• most values are reasonably clustered
• there are no extreme outliers
• every measurement should contribute to the result

Example:

Repeated measurements:

10.1 s, 10.3 s, 10.2 s, 10.4 s, 10.0 s

The mean provides a useful summary because the values are relatively close together.

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When the Median Is Useful

The median is often useful when:

• extreme values are present
• the distribution is skewed
• you want the middle position
• a few unusually high or low values would distort the mean

Examples include:

• household income
• house prices
• travel times
• waiting times

Suppose houses sell for:

$300,000, $320,000, $340,000, $350,000, $2,000,000

The mean is heavily influenced by the $2 million property.

The median:

$340,000

may better describe the centre of these particular sale prices.

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When the Mode Is Useful

The mode is useful when you want to know:

What occurs most frequently?

It is especially useful for:

• categories
• product sizes
• survey responses
• common choices
• frequently occurring values

For example:

T-shirt sizes sold:

M, L, M, S, M, L, XL, M

Mode = M

The mean and median would not be meaningful for these category labels.

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Choosing the Measure Based on the Question

Consider a shoe store.

The manager wants to know:

Which shoe size should we stock most heavily?

The mode is probably useful because it identifies the most commonly purchased size.

Now consider:

What is the middle house price in this neighbourhood?

The median directly answers that question and is less affected by unusually expensive houses.

Now consider:

What was the average reaction time across repeated laboratory trials?

The mean may be appropriate because all numerical measurements contribute.

The correct measure depends on what you want to know.

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Worked Example: House Prices

Suppose six houses sell for:

$280,000

$300,000

$310,000

$330,000

$340,000

$1,200,000

Mean

Total = $2,760,000

Mean = $2,760,000 ÷ 6

Mean = $460,000

Median

The middle values are:

$310,000 and $330,000

Median = ($310,000 + $330,000) ÷ 2

Median = $320,000

The mean suggests $460,000.

But five of the six houses sold for $340,000 or less.

In this situation, the median gives important information about the centre because the expensive house strongly influences the mean.

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Worked Example: Test Scores

Consider:

68, 71, 74, 76, 76, 78, 81

Mean:

524 ÷ 7 ≈ 74.9

Median:

76

Mode:

76

All three values are relatively close.

There are no extreme values dramatically changing the mean.

In this case, all three measures provide useful information, although each describes the dataset differently.

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Worked Example: Product Sales

A shop sells T-shirts in the following sizes:

S, M, M, L, M, XL, L, M, S, M

Which measure should the manager use to determine the most popular size?

Mean?

No. You cannot meaningfully average clothing categories.

Median?

Not usually the most useful choice for this question.

Mode?

Yes.

Mode = M

Therefore, medium is the most frequently sold size.

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The Median Does Not Consider the Size of Every Value

The median depends mainly on position, not on how extreme the largest and smallest values are.

Compare:

Dataset A

1, 5, 6, 7, 10

Median = 6

Dataset B

−500, 5, 6, 7, 10,000

Median = 6

The extreme values changed dramatically.

The median did not change at all.

This resistance to extreme values is both a strength and a limitation.

It makes the median useful when outliers exist, but it also means the median does not reflect the magnitude of every observation.

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The Mode Does Not Have to Be Near the Centre

Consider:

1, 1, 1, 10, 11, 12, 13

Mode = 1

Median = 10

The mode is not necessarily near the mathematical centre of the dataset.

It simply tells us which value occurs most frequently.

This is why you should interpret the word centre carefully when discussing measures of central tendency.

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Frequency Tables

The mode can be particularly easy to identify from a frequency table.

The highest frequency is:

8

This corresponds to:

2 books

Therefore:

Mode = 2 books

The frequency itself is not the mode.

The mode is the value associated with the highest frequency.

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Finding the Median from a Frequency Table

Consider:

There are:

2 + 3 + 4 + 2 = 11 observations

The median is the sixth observation.

Expanded data:

1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4

The sixth value is:

3

Therefore:

Median = 3

You do not always need to write out the entire dataset, but doing so can help while learning.

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Interpreting the Median in Context

Suppose the median waiting time at a clinic is:

18 minutes

A useful interpretation is:

The middle waiting time was 18 minutes when the waiting times were ordered.

It does not mean:

• everyone waited 18 minutes
• the mean was necessarily 18 minutes
• 18 minutes was necessarily the most common waiting time

Always interpret the statistic according to what it actually measures.

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Interpreting the Mode in Context

Suppose the mode of shoe sizes sold is:

Size 9

A useful interpretation is:

Size 9 was the most frequently sold shoe size in the dataset.

It does not mean:

• the average customer wears size 9
• half of customers wear size 9
• size 9 is the middle size

Mode tells us about frequency, not position or equal sharing.

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Comparing Two Datasets

Consider:

Dataset A

4, 5, 5, 6, 10

Mean = 6

Median = 5

Mode = 5

Dataset B

5, 5, 6, 7, 7

Mean = 6

Median = 6

Modes = 5 and 7

Both datasets have the same mean.

But their medians and modes differ.

This demonstrates why looking at multiple measures can reveal information hidden by a single average.

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Outliers and Measures of Centre

Suppose a class records:

12, 13, 14, 14, 15, 16, 60

Mean:

144 ÷ 7 ≈ 20.6

Median:

14

Mode:

14

The value 60 strongly affects the mean.

It does not affect the median nearly as much and does not change the mode.

When outliers are present, compare several measures before deciding how to summarize the data.

Real-World Example: Travel Times

Suppose travel times to school are:

18, 19, 20, 20, 21, 22, 65 minutes

Mean:

185 ÷ 7 ≈ 26.4 minutes

Median:

20 minutes

Mode:

20 minutes

Why was one trip 65 minutes?

Perhaps there was:

• a traffic accident
• severe weather
• a bus delay

If you want to describe a typical journey, the median of 20 minutes may be more informative than the mean of 26.4 minutes.

But the 65-minute trip is still real data and may be important depending on the question.

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No Measure Is Automatically "Correct"

Suppose someone asks:

"What is the average?"

A useful response may be:

"Which average do you mean?"

Mean, median, and mode all describe data differently.

For example, a business may care about:

Mean sale value — useful for revenue calculations.

Median sale value — useful for describing a typical transaction when extreme sales occur.

Mode product — useful for identifying the most commonly purchased item.

All three can be useful simultaneously.

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Common Mistakes

Forgetting to order data before finding the median

Always arrange numerical data before locating the middle.

Choosing one of the middle values when there is an even number

Calculate the mean of the two middle values.

Assuming the median must appear in the dataset

With an even number of observations, it may fall between two values.

Saying the mode is the largest number

Mode means most frequent, not largest.

Saying the mode is the highest frequency

The mode is the value associated with the highest frequency.

Saying the mode is zero when there is no mode

"No mode" and "mode = 0" mean very different things.

Assuming there can only be one mode

Datasets can be bimodal or multimodal.

Assuming every dataset has a mode

Some datasets have no mode.

Assuming the median is always better than the mean

The appropriate measure depends on the data and the question.

Ignoring extreme values

Outliers can strongly affect the mean.

Using the mean with categories

You cannot meaningfully calculate the mean of categories such as colours or favourite subjects.

Assuming equal means imply equal datasets

Datasets can have the same mean while having different medians, modes, ranges, and distributions.

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Key Terms

Measure of centre — A value used to describe the central or typical part of a dataset.

Central tendency — The tendency of data to cluster around a central value.

Mean — The sum of all numerical values divided by the number of values.

Median — The middle value of an ordered dataset.

Mode — The value or category occurring most frequently.

Average — A general term for a value used to represent a dataset; it may refer to mean, median, or mode depending on context.

Ordered data — Data arranged from smallest to largest or largest to smallest.

Frequency — The number of times a value or category occurs.

Frequency table — A table showing values or categories and how often they occur.

Unimodal — Having one mode.

Bimodal — Having two modes.

Multimodal — Having more than two modes.

No mode — A situation in which no value occurs more frequently than the others.

Outlier — A value unusually far from most other observations.

Extreme value — A particularly high or low observation.

Range — The difference between the largest and smallest values.

Distribution — The way values are spread throughout a dataset.

Skewed data — Data whose distribution extends farther in one direction than the other.

Categorical data — Data representing groups or categories rather than numerical measurements.

Numerical data — Data represented using numbers with quantitative meaning.

Representative value — A value used to summarize or describe a dataset.

Robust/Resistant measure — A statistic that is relatively unaffected by extreme values; the median is more resistant to outliers than the mean.

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Key Takeaways

• Mean, median, and mode are measures used to summarize data.
• The median is the middle value of an ordered dataset.
• Always put numerical data in order before finding the median.
• With an odd number of values, the median is the single middle value.
• With an even number of values, calculate the mean of the two middle values.
• The median does not necessarily appear in the original dataset.
• The median is less affected by extreme values than the mean.
• This makes the median useful for datasets containing strong outliers or skewed values.
• The mode is the value or category occurring most frequently.
• A dataset can have one mode, several modes, or no mode.
• The mode can be used with categorical data.
• Mean, median, and mode describe different features of a dataset.
• The mean uses the numerical size of every value.
• The median depends mainly on the position of values.
• The mode depends on frequency.
• The mean is often useful when numerical data contain no extreme values.
• The median is often useful when extreme values strongly influence the mean.
• The mode is particularly useful when you want to identify the most common value or category.
• A higher or lower mean does not tell you everything about a dataset.
• Two datasets can have identical means but different medians, modes, and distributions.
• No single measure of centre is automatically best.
• The appropriate measure depends on both the data and the question being asked.
• Statistics should always be interpreted in their real-world context.

A useful way to remember the three measures is:

Mean → Equal sharing

Median → Middle

Mode → Most common

The important question is not simply:

"Can I calculate all three?"

It is:

"Which measure gives the most useful description of these data, and why?"

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Check Your Understanding

1. Find the median:

3, 7, 5, 9, 4

2. Find the median:

12, 8, 15, 11, 20, 9

3. Find the mode:

4, 7, 4, 8, 6, 4, 9

4. Find the mode:

2, 3, 3, 5, 5, 7

5. Does this dataset have a mode?

1, 3, 5, 7, 9

Explain your answer.

6. Calculate the mean, median, and mode:

3, 4, 4, 5, 9

7. Calculate the mean and median:

8, 9, 10, 11, 100

Which measure better represents the centre of most of these values? Explain.

8. Why must numerical data be ordered before finding the median?

9. Explain why the median is less affected by extreme values than the mean.

10. A neighbourhood has these house prices:

$280,000, $300,000, $315,000, $325,000, $340,000, $2,000,000

Calculate the median.

11. Would the mean or median provide more useful information about the centre of these house prices? Explain using the data.

12. A shoe shop sells these sizes:

7, 8, 9, 8, 10, 8, 9, 11, 8

Find the mode and explain what it means to the shop manager.

13. Why can the mode be used with favourite colours but the mean cannot?

14. A dataset has two values tied for the highest frequency. What is this called?

15. Explain the difference between what the mean, median, and mode represent.

16. The median waiting time at a restaurant is 24 minutes. Explain what this tells you about the dataset.

17. Dataset A has a mean of 15 and Dataset B also has a mean of 15. Give one reason why the two datasets could still be very different.

18. Which measure would probably be most useful for each situation? Explain your choices.

a. Most popular T-shirt size
b. Typical house price when several extremely expensive homes are included
c. Average time from several consistent laboratory trials

19. A dataset has the following frequency table:

Identify the mode.

20. Challenge: Consider:

4, 5, 5, 6, 7, 8, 50

Calculate the mean, median, and mode. Then explain which measure you would choose to represent a typical value in this dataset and justify your choice using the distribution of the data.