Summarizing Data

Collecting data is only the beginning of statistical analysis.

A long list of numbers can be difficult to understand:

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

Statistical measures allow us to turn these raw data into a much clearer description.

For example:

• Mean = 13.8
• Median = 14
• Mode = 14
• Range = 4

These four numbers summarize important features of the entire dataset.

But effective data analysis is not simply a matter of calculating every statistic available. We need to decide which statistics are appropriate, compare them, consider variability and outliers, and explain what the results mean in context.

A good statistical summary should help someone understand the data without needing to inspect every individual value.

From Raw Data to Useful Information

Raw data are the original observations or measurements collected during an investigation.

For example, suppose students measure plant heights:

18, 19, 20, 18, 21, 20, 19, 22 cm

Looking at the individual values gives us information, but a statistical summary makes the overall pattern easier to see.

Mean:

(18 + 19 + 20 + 18 + 21 + 20 + 19 + 22) ÷ 8

= 157 ÷ 8

= 19.625 cm

Median:

Ordered data:

18, 18, 19, 19, 20, 20, 21, 22

Median = (19 + 20) ÷ 2

= 19.5 cm

Mode:

18, 19, and 20 cm

Range:

22 − 18 = 4 cm

Now we know both the centre and the spread of the data.

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Measures of Centre

A measure of centre provides a value that represents the central or typical part of a dataset.

The three measures we have studied are:

• mean
• median
• mode

They describe the centre in different ways.

Choosing between them depends on the data and the question being asked.

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The Mean

The mean uses every numerical value.

Mean = sum of values ÷ number of values

For:

6, 7, 8, 9, 10

Mean = 40 ÷ 5

Mean = 8

The mean is often useful when:

• data are numerical
• all values should contribute
• there are no extreme outliers severely distorting the result
• repeated measurements are being summarized

Because every value contributes, the mean contains a lot of information about the dataset.

However, this also makes it sensitive to outliers.

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The Median

The median is the middle value after the data are placed in numerical order.

For:

5, 7, 8, 9, 12

Median = 8

The median is often useful when:

• outliers are present
• data are strongly skewed
• you want the middle position
• extreme values would make the mean misleading

Unlike the mean, the median is relatively resistant to extreme values.

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The Mode

The mode is the most frequently occurring value or category.

For:

2, 3, 3, 3, 5, 7

Mode = 3

The mode is especially useful when:

• the most common value matters
• the data are categorical
• you are studying popularity or frequency

Examples include:

• most common shoe size
• most popular product
• most frequently chosen answer
• most common species observed
• most common transport method

Unlike the mean and median, the mode can describe non-numerical categories.

Measures of Spread

Knowing the centre is not enough.

Consider:

Dataset A

18, 19, 20, 21, 22

Dataset B

5, 10, 20, 30, 35

Both have:

Mean = 20

But their distributions are very different.

Dataset A is tightly clustered.

Dataset B is much more spread out.

We therefore also need a measure of spread.

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The Range

The simplest measure of spread is the range.

Range = maximum − minimum

For Dataset A:

Range = 22 − 18

Range = 4

For Dataset B:

Range = 35 − 5

Range = 30

Now the difference between the datasets becomes much clearer.

Both have:

Mean = 20

But:

Dataset A: Range = 4

Dataset B: Range = 30

Dataset B shows much greater variability.

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Centre and Spread Should Be Used Together

A strong statistical summary often includes:

a measure of centre + a measure of spread

For example:

Mean = 24.6 cm, range = 3.2 cm

This tells us much more than:

Mean = 24.6 cm

alone.

The mean tells us approximately where the data are centred.

The range tells us how far apart the smallest and largest observations are.

Together they provide a more complete description.

Choosing the Appropriate Measure

The most appropriate statistic depends on:

• the type of data
• the distribution of the data
• whether outliers are present
• the question being investigated

There is no single statistic that is always best.

A good data analyst asks:

What am I trying to describe?

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Choosing the Mean

Suppose a student measures reaction time five times:

12.1 s, 12.2 s, 12.0 s, 12.3 s, 12.2 s

The values are numerical and tightly grouped.

Mean:

60.8 ÷ 5 = 12.16 s

Using the mean makes sense because all five measurements provide useful information and there are no obvious extreme values.

A suitable summary might be:

"The mean reaction time was 12.16 s."

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Choosing the Median

Suppose journey times to school are:

18, 19, 20, 20, 21, 22, 68 minutes

Mean:

188 ÷ 7 ≈ 26.9 minutes

Median:

20 minutes

The unusually long 68-minute journey strongly affects the mean.

If the goal is to describe a typical journey, the median may be more informative.

A suitable statement would be:

"The median journey time was 20 minutes; the mean was raised substantially by one unusually long journey."

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Choosing the Mode

Suppose a shop records T-shirt sizes sold:

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

The question is:

Which size is sold most frequently?

The appropriate statistic is the mode.

Mode = M

A suitable conclusion is:

"Medium was the modal size, meaning it was purchased more frequently than any other size."

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Choosing the Range

Suppose two machines manufacture bolts.

Machine A

49.9, 50.0, 50.1, 50.0, 50.0 mm

Machine B

48.6, 49.3, 50.0, 50.8, 51.4 mm

Machine A:

Range = 50.1 − 49.9

Range = 0.2 mm

Machine B:

Range = 51.4 − 48.6

Range = 2.8 mm

Machine A has much lower variability based on the range.

In manufacturing, that difference could be very important.

A Statistical Summary

A useful statistical summary might include:

• number of observations
• mean
• median
• mode, if useful
• range
• possible outliers
• a short interpretation

For example:

Dataset:

8, 9, 9, 10, 10, 11, 13

Number of observations = 7

Mean = 70 ÷ 7 = 10

Median = 10

Mode = 9 and 10

Range = 13 − 8 = 5

A useful summary is:

"The dataset contains seven observations centred around 10. The mean and median are both 10, while 9 and 10 are the modes. The range is 5, indicating that the values are relatively closely grouped."

That is much more informative than simply writing:

Mean = 10

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Do We Always Need Every Statistic?

No.

Calculating every possible statistic does not automatically produce better analysis.

Suppose favourite colours are:

Blue, Red, Blue, Green, Blue, Red

Calculating a mean would make no sense.

The useful statistic is:

Mode = Blue

Now suppose a scientist collects repeated temperature measurements:

24.8, 24.9, 25.0, 25.1, 25.0 °C

The mode may not be particularly important.

The mean and range may be more useful.

Good statistical analysis involves selection, not just calculation.

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Comparing Multiple Datasets

Statistics become especially powerful when comparing groups.

Suppose two groups of plants receive different fertilizers.

Fertilizer A

18, 19, 20, 20, 21, 22 cm

Fertilizer B

13, 16, 19, 21, 24, 27 cm

Fertilizer A

Mean = 120 ÷ 6

Mean = 20 cm

Range = 22 − 18

Range = 4 cm

Fertilizer B

Mean = 120 ÷ 6

Mean = 20 cm

Range = 27 − 13

Range = 14 cm

Both groups have exactly the same mean.

But Fertilizer B produced much greater variability.

A strong comparison would say:

"Both groups had a mean plant height of 20 cm, but Fertilizer B had a much larger range (14 cm compared with 4 cm), indicating greater variability in plant height."

This is more informative than saying:

"Both fertilizers produced the same results."

They did not.

They produced the same mean, but different distributions.

Compare Like with Like

When comparing datasets, use the same statistical measures.

For example, avoid comparing:

Group A mean = 75

with:

Group B median = 82

unless there is a clear reason.

A more useful comparison is:

Now the statistics can be compared directly.

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Comparing Centre

Suppose:

Class A

Mean score = 74%

Class B

Mean score = 81%

We can say:

"Class B had a higher mean score than Class A by 7 percentage points."

Notice the language.

We are not saying:

"Every student in Class B performed better."

A group statistic describes the group as a whole, not every individual member.

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Comparing Spread

Suppose:

Class A

Range = 12 percentage points

Class B

Range = 37 percentage points

We can say:

"Class B had a larger range, indicating greater spread between its highest and lowest scores."

Or:

"Class A's results were more tightly grouped based on the range."

These statements describe the data precisely.

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Comparing Centre and Spread Together

Suppose:

A useful interpretation is:

"The classes had the same mean score of 76%, but Class B had a much larger range. This indicates that Class B's scores were more variable even though the average performance was the same."

Notice that we have combined:

centre + spread + interpretation

This is the basis of a strong statistical comparison.

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Outliers and Statistical Summaries

Before summarizing data, check for possible outliers.

Consider:

10, 11, 11, 12, 12, 13, 64

Mean:

133 ÷ 7 = 19

Median:

12

Range:

64 − 10 = 54

The mean of 19 gives a very different impression from the median of 12.

The reason is the extreme value:

64

A good summary should mention this.

For example:

"The median was 12, while the mean was 19 because the unusually high value of 64 pulled the mean upward. The outlier also produced a large range of 54."

Summarizing Data With an Outlier

Suppose:

20, 21, 21, 22, 22, 23, 70

Mean:

199 ÷ 7 ≈ 28.4

Median:

22

Mode:

21 and 22

Range:

70 − 20 = 50

A weak summary would say:

"The average is 28.4."

A stronger summary would say:

"Most observations lie between 20 and 23, with a median of 22. The value 70 is a possible outlier that raises the mean to 28.4 and increases the range to 50."

The second statement gives the reader a much more accurate picture.

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Statistics Need Context

Suppose you calculate:

Mean = 18.6

That number means very little by itself.

18.6 what?

It could represent:

• centimetres
• seconds
• kilograms
• degrees Celsius
• points
• dollars

A statistical conclusion should include:

• what was measured
• the statistic
• the value
• the units
• what the result suggests

For example:

"The plants had a mean height of 18.6 cm."

is much clearer than:

"The mean was 18.6."

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Statistical Language

Using accurate language helps prevent misleading conclusions.

Instead of:

"Group A was better."

write:

"Group A had a higher mean test score."

Instead of:

"Group B was all over the place."

write:

"Group B showed greater variability, with a range of 24 compared with 8 for Group A."

Instead of:

"The weird number ruined the average."

write:

"The unusually high value increased the mean and range."

Statistical language should be:

• specific
• measurable
• neutral
• evidence-based

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Useful Statistical Sentence Starters

When describing centre:

"The mean _____ was..."

"The median value was..."

"The modal category was..."

When describing spread:

"The data had a range of..."

"Group A showed greater variability than Group B..."

"The values were relatively tightly clustered..."

When comparing:

"Group A had a higher mean than Group B..."

"Although the groups had similar medians, their ranges differed..."

"Both groups had the same mean, but..."

When discussing outliers:

"The unusually high value of _____ increased the mean..."

"The median may provide a more representative measure of centre because..."

When concluding:

"These results suggest..."

"Based on the mean and range..."

"The statistical summary indicates..."

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Avoiding Overstatements

Statistics describe evidence.

They do not always prove why something happened.

Suppose plants receiving Fertilizer A have a higher mean height than plants receiving Fertilizer B.

You may be able to say:

"Plants receiving Fertilizer A had a higher mean height in this investigation."

But be careful with:

"Fertilizer A always makes plants grow taller."

The experiment may have involved:

• a small sample
• uncontrolled variables
• measurement uncertainty
• natural biological variation

Good statistical conclusions remain within what the evidence actually supports.

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Sample Size Matters

Compare:

Experiment A

Three plants:

10, 11, 30 cm

Mean = 17 cm

Experiment B

Three hundred plants:

Mean = 17 cm

The means are identical.

But the second result is based on far more observations.

This does not automatically make every aspect of Experiment B perfect, but sample size is important when evaluating statistical evidence.

A useful summary may therefore include:

n = number of observations

For example:

Mean height = 17 cm (n = 300)

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Statistics Do Not Replace the Original Data

A statistical summary simplifies data.

That is useful—but simplification means some information is lost.

Consider:

Dataset A

4, 5, 5, 5, 6

Dataset B

1, 4, 5, 6, 9

Both have:

Mean = 5

Median = 5

But:

Dataset A range = 2

Dataset B range = 8

Even mean and median together do not tell the whole story.

Graphs and original data can reveal patterns that summary statistics hide.

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Statistics and Graphs Work Together

A strong data analysis often combines:

statistical measures + graphical displays

Useful graphs include:

• bar graphs
• histograms
• dot plots
• box plots
• scatter plots
• line graphs

Statistics summarize numerical features.

Graphs help reveal:

• patterns
• clusters
• gaps
• trends
• outliers
• differences between groups

Neither should automatically replace the other.

A Step-by-Step Strategy for Summarizing Data

When given a dataset, use the following process.

Examine the data

Ask:

• What is being measured?
• What are the units?
• How many observations are there?
• Are the data numerical or categorical?

Look at the distribution

Ask:

• Are values clustered?
• Are they widely spread?
• Are there obvious gaps?
• Are there possible outliers?

Choose a measure of centre

Consider:

Mean — useful when all numerical values should contribute and strong outliers are absent.

Median — useful when extreme values or skewed data could distort the mean.

Mode — useful when the most common value or category matters.

Choose a measure of spread

At this level:

Range = maximum − minimum

Compare statistics

If multiple datasets are present, compare:

• centre
• spread
• outliers
• patterns

Interpret

Explain what the statistics mean in the actual situation.

Conclude

Use evidence from the data to answer the original question.

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Worked Example: Experimental Data

Students investigate two ramp surfaces.

They measure the time a toy car takes to travel a fixed distance.

Surface A

2.1, 2.2, 2.2, 2.3, 2.2 s

Mean:

11.0 ÷ 5 = 2.2 s

Median = 2.2 s

Mode = 2.2 s

Range:

2.3 − 2.1 = 0.2 s

Surface B

2.0, 2.1, 2.4, 2.7, 2.8 s

Mean:

12.0 ÷ 5 = 2.4 s

Median = 2.4 s

Range:

2.8 − 2.0 = 0.8 s

A suitable summary is:

"The car travelled the measured distance more quickly on Surface A on average, with a mean time of 2.2 s compared with 2.4 s on Surface B. Surface A also produced less variable results, with a range of 0.2 s compared with 0.8 s."

This conclusion uses both:

centre and spread

Worked Example: Choosing Median Instead of Mean

Suppose delivery times are:

18, 19, 20, 20, 21, 22, 85 minutes

Mean:

205 ÷ 7 ≈ 29.3 minutes

Median:

20 minutes

Range:

85 − 18 = 67 minutes

A strong summary is:

"The median delivery time was 20 minutes. One unusually long delivery of 85 minutes increased the mean to approximately 29.3 minutes and produced a range of 67 minutes. The median therefore provides useful information about the centre of most of these delivery times."

Notice that we did not hide the outlier.

We explained its effect.

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Worked Example: Comparing Three Groups

Suppose three classes complete the same test.

We can make several evidence-based observations.

Class A and Class B have the same mean, but Class B has a much larger range.

Therefore, Class B's scores show greater variability based on the range.

Class C has the highest mean and median of the three groups.

Its range is similar to Class A's and much smaller than Class B's.

A useful summary would be:

"Classes A and B both had mean scores of 78%, but Class B showed considerably greater variability, with a range of 38 compared with 12 for Class A. Class C had a mean and median of 84% and a range of 14."

Notice that the conclusion reports exactly what the statistics show without making claims about individual students.

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Statistical Summaries in Science

Scientists use statistical summaries to:

• summarize repeated measurements
• compare experimental treatments
• describe populations
• evaluate consistency
• identify unusual observations
• communicate evidence
• support conclusions

Suppose an investigation asks:

Does increasing fertilizer concentration affect plant growth?

A scientist might report:

The medium and high treatments have similar means.

But the high treatment has much greater variability.

That difference might be scientifically important and deserves discussion.

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From Calculation to Interpretation

Consider:

Mean = 12.4 cm

Range = 1.2 cm

Calculation tells us:

what the numbers are

Interpretation tells us:

what the numbers mean

For example:

"The plants had a mean height of 12.4 cm and a relatively small range of 1.2 cm, indicating that the measured heights were closely grouped."

This is much stronger scientific communication.

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Questions to Ask Before Drawing a Conclusion

Before writing your final interpretation, ask:

What does the measure of centre tell me?

What does the measure of spread tell me?

Are there any outliers?

Would an outlier make the mean misleading?

How large is the sample?

Are the groups being compared fairly?

Do the statistics support my conclusion?

Am I claiming more than the evidence shows?

These questions turn calculations into statistical thinking.

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Common Mistakes

Reporting only the mean

Centre alone does not describe variability.

Calculating every statistic without considering usefulness

Choose statistics that answer the question.

Using the mean for categorical data

A mean cannot meaningfully summarize categories such as colours or brands.

Ignoring outliers

Extreme values can strongly affect the mean and range.

Automatically deleting outliers

Investigate them first.

Comparing different measures

Avoid comparing the mean of one group directly with the median of another unless there is a clear reason.

Assuming equal means mean equal datasets

Datasets can have the same mean and very different spreads.

Confusing range with maximum

Range = maximum − minimum.

Forgetting units

Statistical values should normally include the appropriate units.

Using vague language

Replace:

"Group A was better."

with a measurable statement such as:

"Group A had a higher mean score."

Making conclusions about individuals from group averages

A group mean does not describe every member of the group.

Claiming causation from a simple comparison

A statistical difference does not automatically prove what caused it.

Ignoring sample size

A statistic based on very few observations may provide limited information about a larger population.

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Key Terms

Data — Information collected through measurements, observations, surveys, or experiments.

Raw data — Original observations before they are summarized or processed.

Dataset — A collection of related data values.

Statistical summary — A concise description of important features of a dataset using statistical measures.

Statistic — A numerical value calculated from data.

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

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

Median — The middle value of ordered data.

Mode — The most frequently occurring value or category.

Measure of spread — A statistic describing how much values vary.

Range — The difference between the maximum and minimum values.

Maximum — The largest observation.

Minimum — The smallest observation.

Variability — The amount of difference or spread among observations.

Variation — Differences among values in a dataset.

Distribution — The overall arrangement and spread of data values.

Outlier — A value unusually far from most other observations.

Anomaly — A result that does not fit the general or expected pattern.

Cluster — A group of observations located relatively close together.

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

Sample size — The number of observations in a sample.

Population — The entire group about which conclusions may be made.

Categorical data — Data representing categories or labels.

Numerical data — Data represented by meaningful numerical values.

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

Statistical comparison — The use of statistics to identify similarities and differences between datasets.

Interpretation — An explanation of what calculated statistics mean in context.

Evidence-based conclusion — A conclusion supported by observations, measurements, and appropriate analysis.

Statistical language — Precise terminology used to describe and compare data.

Sample size — The number of observations included in a sample or dataset.

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Key Takeaways

• Raw data become easier to understand when summarized using appropriate statistics.
• Measures of centre describe the central or typical part of a dataset.
• The main measures of centre are mean, median, and mode.
• The mean uses every numerical observation.
• The median describes the middle position of ordered data.
• The mode identifies the most frequently occurring value or category.
• The most appropriate measure depends on the type and distribution of the data.
• The mean is sensitive to extreme values.
• The median is more resistant to outliers.
• The mode is particularly useful for categorical data.
• Measures of spread describe variability.
• The range is calculated using maximum − minimum.
• A good statistical summary often combines a measure of centre with a measure of spread.
• Two datasets can have the same mean but very different variability.
• Equal averages do not mean that datasets are identical.
• Outliers can strongly affect the mean and range.
• Outliers should be investigated rather than automatically removed.
• Comparing datasets requires using equivalent statistics where possible.
• Statistical conclusions should include units and context.
• Precise statistical language is better than vague descriptions such as "better" or "worse."
• Group statistics describe groups, not every individual observation.
• Sample size should be considered when evaluating evidence.
• Statistics simplify data, so examining graphs and original values can still be important.
• Graphs and statistics complement each other.
• Statistical differences do not automatically establish causation.
• Strong conclusions are supported by the data and do not claim more than the evidence demonstrates.
• Good data analysis involves calculation, comparison, interpretation, and communication.

A useful summary strategy is:

Examine → Choose → Calculate → Compare → Interpret → Conclude

And whenever you summarize data, ask:

What is the centre?

How much do the values vary?

Are there any outliers?

Which statistics best answer the question?

What do these numbers mean in context?

What conclusion is actually supported by the evidence?

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Check Your Understanding

1. Calculate the mean, median, mode, and range:

5, 6, 6, 7, 8, 10

2. Which statistic in Question 1 describes the spread of the data?

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

4. Which measure of centre would be most useful for finding the most popular shoe size? Explain.

5. Which measure might be useful for describing typical house prices when a few extremely expensive houses are included? Explain.

6. Two classes have the following statistics:

Class A: Mean = 78%, Range = 10

Class B: Mean = 78%, Range = 34

What can you conclude?

7. Why would it be incorrect to say that the two classes in Question 6 have identical results?

8. Dataset A:

18, 19, 20, 21, 22

Dataset B:

5, 10, 20, 30, 35

Calculate the mean and range of each.

9. Compare the datasets in Question 8 using appropriate statistical language.

10. Consider:

12, 13, 13, 14, 15, 60

Calculate the mean and median.

11. Which measure in Question 10 gives a more useful description of where most of the values are located? Explain.

12. How does the outlier in Question 10 affect the range?

13. Why should a statistical summary usually include context and units?

14. Rewrite this statement using more precise statistical language:

"Group B did way better than Group A."

Assume Group B had a mean score of 84% and Group A had a mean score of 76%.

15. A scientist obtains:

Treatment A: Mean = 16.2 cm, Range = 1.8 cm

Treatment B: Mean = 16.4 cm, Range = 8.7 cm

Compare the two treatments.

16. Why might a scientist examine a graph as well as calculating summary statistics?

17. Explain why a higher mean in one experimental group does not automatically prove that the treatment caused the difference.

18. A survey asks students for their favourite sport. Which measure of centre is appropriate, and why?

19. Write a statistical summary of:

21, 22, 22, 23, 23, 24, 25

Include at least one measure of centre and one measure of spread.

20. Challenge: Two experimental groups produce:

Group A:

9, 10, 10, 11, 10, 10

Group B:

6, 7, 10, 12, 14, 11

Calculate the mean, median, and range for both groups. Then write a short scientific paragraph comparing their centres and variability and explaining what conclusions the statistical evidence supports.