Range and Variability

When we analyze data, knowing the centre of a dataset is useful—but it does not tell us the whole story.

Consider these two datasets:

Dataset A: 8, 9, 10, 11, 12

Dataset B: 2, 6, 10, 14, 18

Both have exactly the same mean:

Mean = 10

But the datasets are clearly different.

Dataset A is tightly clustered around 10.

Dataset B is much more spread out.

This difference is called variability.

Variability describes how much the values in a dataset differ from one another. Measures of spread, such as the range, help us describe this variability numerically.

Understanding variability is essential because two datasets with the same mean or median can represent very different situations.

What Is Variability?

Variability describes the amount of difference or spread among values in a dataset.

Consider:

10, 10, 10, 10, 10

There is no variability.

Every value is identical.

Now consider:

8, 9, 10, 11, 12

There is some variability.

Now:

−10, 0, 10, 20, 30

There is much greater variability.

As the observations become more spread out, variability generally increases.

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Why Variability Matters

Suppose two students each have an average quiz score of 80%.

Student A

78%, 80%, 81%, 79%, 82%

Student B

50%, 65%, 80%, 95%, 110%

Ignoring the unusual grading scale in the example, both could have the same mean, but their performances are very different.

Student A is relatively consistent.

Student B's results vary considerably.

If we looked only at the mean, we would miss this important information.

This is why data analysis often considers:

centre + spread

rather than centre alone.

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What Is the Range?

The range is one of the simplest measures of spread.

Range = highest value − lowest value

For example:

4, 7, 9, 10, 13

Highest value = 13

Lowest value = 4

Range = 13 − 4

Range = 9

This tells us that there are 9 units between the smallest and largest values.

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Calculating the Range

A useful routine is:

Find the highest → Find the lowest → Subtract

Consider:

18, 12, 25, 16, 20

It may help to order the data:

12, 16, 18, 20, 25

Highest = 25

Lowest = 12

Range = 25 − 12

Range = 13

Unlike finding the median, ordering the data is not strictly necessary.

But it makes identifying the minimum and maximum much easier.

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Worked Example: Test Scores

Five students receive:

72, 81, 68, 90, 76

Lowest score = 68

Highest score = 90

Range = 90 − 68

Range = 22 percentage points

This tells us how far apart the highest and lowest scores are.

Worked Example: Temperature

Suppose temperatures recorded during one day are:

24 °C, 27 °C, 30 °C, 32 °C, 29 °C, 25 °C

Lowest = 24 °C

Highest = 32 °C

Range = 32 − 24

Range = 8 °C

The temperatures varied across an interval of 8 °C.

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Range with Negative Numbers

Be careful when the lowest value is negative.

Consider:

−8, −3, 2, 5, 9

Highest = 9

Lowest = −8

Range = 9 − (−8)

Range = 9 + 8

Range = 17

A common mistake is calculating:

9 − 8 = 1

Remember that the lowest value is negative eight, not positive eight.

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Range Can Be Zero

Consider:

7, 7, 7, 7, 7

Highest = 7

Lowest = 7

Range = 7 − 7

Range = 0

A range of zero means there is no spread between the highest and lowest values.

Every observation is the same.

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Comparing Datasets Using Range

Consider:

Dataset A

18, 19, 20, 21, 22

Mean = 20

Range = 22 − 18

Range = 4

Dataset B

5, 10, 20, 30, 35

Mean = 20

Range = 35 − 5

Range = 30

Both datasets have the same mean.

But Dataset B has a much larger range.

We can say:

Dataset B shows greater variability based on the range.

Centre and Spread Work Together

Measures of centre include:

• mean
• median
• mode

Measures of spread describe how much values vary.

At this level, the main measure of spread we are using is:

range

Suppose:

Class A

Mean = 75%

Range = 8 percentage points

Class B

Mean = 75%

Range = 42 percentage points

The classes have the same mean.

But Class A's results are much more tightly grouped.

Class B's results show much greater variability.

Reporting both statistics gives a much clearer picture:

Class A: mean = 75%, range = 8

Class B: mean = 75%, range = 42

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Low Variability

A dataset with low variability has values relatively close together.

For example:

48, 49, 50, 50, 51, 52

The results are clustered around approximately 50.

Low variability can indicate:

• consistent measurements
• similar individuals
• stable conditions
• predictable results

However, low variability is not automatically "good."

Whether it is desirable depends on the situation.

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High Variability

A dataset with high variability contains values that are more widely spread.

For example:

20, 31, 48, 63, 79

High variability might result from:

• natural differences
• inconsistent measurements
• changing conditions
• uncontrolled variables
• different groups within the sample
• measurement error

High variability is not automatically "bad."

It is information that needs to be explained and interpreted.

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Variability in Scientific Experiments

Suppose a student measures the time required for a pendulum to complete ten oscillations.

Experiment A

12.1 s
12.2 s
12.0 s
12.1 s
12.2 s

Range:

12.2 − 12.0 = 0.2 s

Experiment B

10.8 s
11.7 s
12.1 s
12.9 s
13.5 s

Range:

13.5 − 10.8 = 2.7 s

Experiment B shows much greater variability.

That should make the scientist ask:

Why?

Possible explanations include:

• inconsistent release position
• timing errors
• changing pendulum length
• counting mistakes
• changing experimental conditions

Variability Is Not Always Experimental Error

Suppose you measure the heights of 100 students.

The heights will vary.

That does not mean the measurements are necessarily poor.

Humans naturally differ in height.

This is called natural variation.

Biological data often show substantial natural variability because organisms differ genetically and environmentally.

Examples include:

• height
• mass
• heart rate
• plant growth
• seed germination
• reaction time
• lifespan

Scientists need to distinguish between:

variation that naturally exists

and

variation introduced by the measurement or experimental method.

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Variability and Reliability

Repeated measurements that are close together often suggest good repeatability.

Suppose a balance produces:

25.1 g

25.0 g

25.1 g

25.0 g

25.1 g

The range is:

25.1 − 25.0 = 0.1 g

The measurements are highly consistent.

Now suppose another method produces:

23.8 g

25.7 g

24.1 g

26.2 g

25.0 g

Range:

26.2 − 23.8 = 2.4 g

These measurements show much greater variation.

Greater variability in repeated measurements can reduce our confidence in a single reported value.

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Consistency Does Not Guarantee Accuracy

Suppose the true mass of an object is:

50.0 g

A faulty balance gives:

53.1 g, 53.0 g, 53.1 g, 53.0 g, 53.1 g

These measurements have a very small range.

They are consistent.

But they are also inaccurate.

A systematic error has shifted all the measurements upward.

Therefore:

Low variability can indicate precision or consistency, but it does not automatically guarantee accuracy.

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Outliers Can Strongly Affect the Range

Consider:

10, 11, 12, 12, 13

Range:

13 − 10 = 3

Now change the final value:

10, 11, 12, 12, 50

Range:

50 − 10 = 40

One extreme value has increased the range dramatically.

This is an important limitation of the range:

It depends only on the smallest and largest values.

The Range Ignores Most of the Data

Consider:

Dataset A

1, 2, 3, 4, 10

Range = 9

Dataset B

1, 9, 9, 9, 10

Range = 9

Both datasets have exactly the same range.

But the values between the minimum and maximum are arranged very differently.

The range uses only:

minimum and maximum

It ignores the positions of every other value.

Therefore, the range is useful but incomplete.

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Same Range, Different Distribution

Consider:

Dataset A

0, 5, 5, 5, 10

Range = 10

Dataset B

0, 1, 5, 9, 10

Range = 10

The ranges are identical.

But Dataset A is strongly concentrated around 5.

Dataset B is more spread across the interval.

The range alone cannot show this difference.

This is why more advanced statistics use other measures of spread.

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Other Measures of Spread

The range is only one way of measuring variability.

Other measures include:

Interquartile range (IQR) — describes the spread of the middle 50% of data.

Variance — measures how far values tend to be from the mean using squared differences.

Standard deviation — describes the typical distance of observations from the mean.

You do not need these measures to understand the basic idea of variability.

The important principle is:

A measure of centre tells us where data are located. A measure of spread tells us how much the data vary.

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Variability and Sample Size

Imagine measuring the heights of three students:

158 cm, 160 cm, 162 cm

Range = 4 cm

You might conclude that student heights show little variability.

But then you measure 300 students and obtain:

145 cm to 191 cm

Range = 46 cm

The first sample simply may not have captured the full variation in the population.

When interpreting variability, consider:

• sample size
• how the sample was selected
• whether the sample represents the population

A range from a very small sample may underestimate the variability that exists in a larger population.

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Variability in Manufacturing

Imagine a factory producing metal rods that should each be:

100.0 mm long

Machine A

99.9, 100.0, 100.1, 100.0, 100.0 mm

Range = 0.2 mm

Machine B

98.7, 99.4, 100.2, 101.0, 100.7 mm

Range = 2.3 mm

Machine A produces much more consistent dimensions.

In manufacturing, low variability is often important because products need to meet precise specifications.

Variability in Sports

Suppose two basketball players both average:

20 points per game

Player A

18, 19, 20, 21, 22

Range = 4

Player B

5, 12, 20, 28, 35

Range = 30

Both players have the same mean.

But Player A is much more consistent from game to game.

Player B has much greater variability.

Depending on the question, this difference could be very important.

If a coach wants predictable scoring, knowing only the mean would not provide enough information.

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Variability in Weather

Suppose two cities both have an average daily temperature of:

20 °C

City A

18, 19, 20, 21, 22 °C

Range = 4 °C

City B

5, 12, 20, 28, 35 °C

Range = 30 °C

Their mean temperatures are identical.

But their weather patterns are dramatically different.

This demonstrates why an average alone may give an incomplete picture of real-world data.

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Variability in Finance

Imagine two investments both have an average annual return of 6%.

Investment A produces returns close to 6% each year.

Investment B sometimes produces large gains and sometimes large losses.

The same average return does not mean the investments behave in the same way.

Their variability differs.

In many real-world situations, variability can be just as important as the average.

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Variability and Prediction

Low variability can sometimes make predictions easier.

Suppose a bus usually arrives after:

9, 10, 10, 10, 11 minutes

You can make a reasonably confident estimate about tomorrow's waiting time.

Now suppose another bus arrives after:

2, 5, 10, 17, 26 minutes

Predicting the next waiting time is much more difficult.

Greater variability often means greater uncertainty when making predictions.

However, prediction also depends on whether the underlying conditions remain similar.

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Comparing Groups Properly

Suppose two fertilizers are tested.

Fertilizer A

Plant heights:

19, 20, 20, 21, 20 cm

Mean = 20 cm

Range = 2 cm

Fertilizer B

Plant heights:

10, 15, 20, 25, 30 cm

Mean = 20 cm

Range = 20 cm

If you reported only:

"Both fertilizers produced a mean height of 20 cm"

you would miss an important difference.

Fertilizer B produced much greater variability.

That might lead to further questions:

• Why are the results so different?
• Were the plants genetically different?
• Were environmental conditions controlled?
• Did plants respond differently to the treatment?
• Was there measurement error?

Variability often reveals scientific questions that averages hide.

Variability Can Reveal Hidden Groups

Suppose a dataset contains:

10, 11, 12, 48, 49, 50

The range is large:

50 − 10 = 40

But looking at the data reveals something interesting.

There appear to be two clusters:

10, 11, 12

and

48, 49, 50

This might suggest that the sample contains two different groups.

Perhaps:

• two species were measured
• two treatments were used
• two age groups were combined
• measurements came from different conditions

A large range may therefore encourage scientists to examine the structure of the dataset more carefully.

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Variability and Experimental Design

If repeated measurements show unexpectedly high variability, ask:

Is the method consistent?

Are control variables actually controlled?

Is the equipment precise enough?

Is the sample naturally variable?

Are there anomalous values?

Do I need more trials?

Variability is not simply something to calculate after an experiment.

It can help scientists evaluate and improve the investigation itself.

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Worked Example: Comparing Two Experiments

A student investigates how long a chemical reaction takes.

Method A

41 s, 42 s, 40 s, 41 s, 42 s

Mean = 41.2 s

Range = 42 − 40

Range = 2 s

Method B

32 s, 37 s, 42 s, 46 s, 49 s

Mean = 41.2 s

Range = 49 − 32

Range = 17 s

The means are identical.

But Method A produces much more consistent measurements.

A scientist should investigate why Method B produces so much more variation.

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Worked Example: Effect of an Outlier

Consider:

15, 16, 16, 17, 18

Range:

18 − 15 = 3

Suppose one measurement is incorrectly recorded as 81 instead of 18:

15, 16, 16, 17, 81

Range:

81 − 15 = 66

The range changes from:

3 to 66

This demonstrates how strongly an outlier or recording error can affect the range.

Before interpreting a very large range, inspect the dataset.

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Interpreting Variability in Context

Never stop at:

"The range is 20."

Ask:

20 what?

and:

What does that mean in this situation?

For example:

"The plant heights had a range of 20 cm, indicating a large difference between the shortest and tallest plants."

Or:

"The repeated reaction times had a range of only 0.4 s, indicating relatively consistent measurements."

Statistics become useful when they are connected to the real-world meaning of the data.

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Common Mistakes

Adding the highest and lowest values

Range requires subtraction:

Range = maximum − minimum

Subtracting in the wrong direction

Always calculate:

highest − lowest

Ignoring negative numbers

Remember:

8 − (−4) = 12

not 4.

Confusing range with mean

The mean describes centre.

The range describes spread.

Saying the range is the largest value

The range is the difference between maximum and minimum.

Assuming a large range automatically means poor data

The variability may be genuine.

Assuming a small range guarantees accurate measurements

Measurements can be consistent but systematically wrong.

Ignoring outliers

One extreme value can dramatically change the range.

Comparing only means

Equal means do not imply equal variability.

Comparing only ranges

Equal ranges do not imply identical distributions.

Forgetting units

If heights are measured in centimetres, the range should also be reported in centimetres.

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Key Terms

Variability — The amount of difference or spread among values in a dataset.

Variation — Differences between observations or measurements.

Spread — The extent to which values are dispersed across a dataset.

Range — The difference between the maximum and minimum values.

Maximum — The largest value in a dataset.

Minimum — The smallest value in a dataset.

Measure of spread — A statistic used to describe variability in data.

Measure of centre — A statistic describing the central or typical value of a dataset.

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

Median — The middle value of ordered data.

Distribution — The way values are arranged or spread throughout a dataset.

Consistency — The degree to which repeated results are similar.

Natural variation — Differences that occur naturally among organisms, materials, or systems.

Measurement variation — Differences between measurements caused by limitations or inconsistencies in measurement.

Outlier — A value unusually far from most of the other observations.

Extreme value — A particularly high or low observation.

Cluster — A group of data values located relatively close together.

Sample — A subset of a larger population that is measured or studied.

Sample size — The number of observations in a sample.

Population — The entire group being investigated.

Repeatability — The closeness of repeated measurements made using the same method under similar conditions.

Accuracy — How close a measurement is to the accepted or true value.

Precision — How closely repeated measurements agree with one another.

Interquartile range (IQR) — The spread of the middle 50% of an ordered dataset.

Variance — A statistical measure based on squared distances from the mean.

Standard deviation — A measure describing how far observations typically lie from the mean.

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Key Takeaways

• Measures of centre do not tell us everything about a dataset.
• Variability describes how much values differ or spread out.
• The range is a simple measure of variability.
• Calculate range using:

Range = highest value − lowest value

• A range of zero means the maximum and minimum are identical.
• Be careful when calculating ranges involving negative numbers.
• A small range indicates that the minimum and maximum are relatively close together.
• A large range indicates that the minimum and maximum are farther apart.
• Two datasets can have the same mean but very different ranges.
• Comparing centre and spread together provides more information than comparing averages alone.
• Low variability means values are relatively clustered.
• High variability means values are more widely spread.
• High variability is not automatically evidence of a poor experiment.
• Biological systems often show natural variation.
• Unexpected variability in repeated measurements can reveal problems with experimental methods or uncontrolled variables.
• Low variability can indicate consistent measurements but does not guarantee accuracy.
• The range is strongly affected by outliers.
• The range uses only the minimum and maximum values.
• Two datasets can have the same range but very different distributions.
• Sample size can affect the observed amount of variability.
• Variability is important in science, manufacturing, weather, sports, finance, and many other real-world situations.
• Greater variability can make predictions less certain.
• Examining variability can reveal clusters or hidden groups in data.
• Scientists use variability to evaluate the quality and meaning of evidence.
• More advanced measures of spread include the interquartile range, variance, and standard deviation.
• Statistics should always be interpreted in the context of the data.

When analyzing a dataset, do not ask only:

"What is the average?"

Also ask:

"How much do the values vary?"

"Are the results tightly clustered or widely spread?"

"Are there extreme values?"

"Is the variability natural, or could it indicate a problem with the investigation?"

"What does this variability mean in the real-world situation?"

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Check Your Understanding

1. Calculate the range:

4, 7, 9, 12, 15

2. Calculate the range:

18, 11, 24, 16, 20

3. Calculate the range:

−6, −2, 3, 8, 11

4. What does a range of zero tell you about a dataset?

5. Explain the difference between a measure of centre and a measure of spread.

6. Consider:

Dataset A: 8, 9, 10, 11, 12

Dataset B: 2, 6, 10, 14, 18

Calculate the mean and range of each dataset.

7. Which dataset in Question 6 has greater variability? Explain.

8. Two classes both have a mean test score of 80%. Class A has a range of 10 percentage points. Class B has a range of 45 percentage points. What does this tell you?

9. Explain why two datasets with the same mean can still be very different.

10. A student records:

5.1 s, 5.2 s, 5.0 s, 5.1 s, 5.2 s

Calculate the range and describe the variability.

11. Another student records:

3.8 s, 4.5 s, 5.1 s, 5.9 s, 6.4 s

Calculate the range and compare it with Question 10.

12. Why might high variability occur in a biological investigation even if the experiment is conducted carefully?

13. Explain why a small range does not necessarily mean measurements are accurate.

14. Consider:

10, 11, 12, 13, 60

Calculate the range.

How does the value 60 affect your interpretation?

15. Why is the range particularly sensitive to outliers?

16. Two datasets both have a range of 20. Does this mean their distributions must be similar? Explain.

17. A factory produces components with the following lengths:

Machine A: 49.9, 50.0, 50.0, 50.1, 50.0 mm

Machine B: 48.5, 49.4, 50.0, 50.8, 51.5 mm

Calculate the range for each machine and compare their consistency.

18. Explain why scientists should consider both the mean and variability when comparing experimental groups.

19. A set of measurements suddenly shows much greater variability than previous trials. Give three possible explanations a scientist should investigate.

20. Challenge: Two fertilizers both produce a mean plant height of 25 cm.

Fertilizer A:

23, 24, 25, 26, 27 cm

Fertilizer B:

10, 18, 25, 32, 40 cm

Calculate the range of each dataset. Then explain why reporting only the mean would give an incomplete picture of the results.