Chapter Overview

Statistics helps us collect, organise and interpret data.

A good statistical question expects variation in its answers.

For example:

How many minutes do students in our class take to reach school?

Different students may give different answers, so data is needed.

1. Statistical Questions

Examples:

• How tall are the students in Class 7?
• How many books did students read last month?
• How much time do students spend travelling to school?

A question such as “How old am I?” is not statistical because it asks for one fixed answer.

2. Arithmetic Mean

The arithmetic mean is a commonly used representative value.

3. Median

For an odd number of observations:

the median is:

For an even number of observations, take the mean of the two middle values.

4. Outliers

Consider:

The value 50 is an outlier.

5. Spread and Range

For:

6. Dot Plots

A dot plot places one dot for each observation above its value.

It helps us see:

• common values;
• clusters;
• gaps;
• unusual values;
• overall spread.

7. Clustered Bar Graphs

Clustered bar graphs place related bars side by side.

They are useful for comparing categories such as:

• boys and girls;
• two different years;
• two teams;
• different classes.

8. What Makes a Question Statistical?

A statistical question expects variation in the answers and is answered by collecting data.

For example, “How tall are the students in Class 7?” is statistical because students have different heights. “How tall is this particular desk?” is not statistical in the same way because it asks for one fixed measurement.

9. Mean and Median Tell Different Stories

The mean uses every value. The median depends on the middle position after arranging the data.

Consider:

The median is 5. The mean is:

The value 20 pulls the mean upward. This shows why an outlier can affect the mean much more than the median.

10. Range and Spread

Range gives a simple measure of how spread out a data set is.

If scores are $12,15,16,17,20$, then:

A larger range usually indicates greater spread, but it depends only on the two extreme values.

11. Reading Dot Plots

A dot plot places one dot for each observation above its value. It allows us to see:

• the most common values;
• clusters;
• gaps;
• unusual values;
• overall spread.

When reading a dot plot, count dots carefully. Several dots may be stacked above the same value.

12. Clustered Bar Graphs

Clustered bar graphs compare two or more categories for each group. Always read:

1. the graph title;
2. horizontal-axis labels;
3. vertical-axis scale;
4. legend or key;
5. units.

13. Choosing a Useful Summary

There is no single statistic that tells everything about a data set. Mean is useful when values are reasonably balanced. Median is often more representative when extreme values are present. Range gives a quick sense of spread.

14. Complete Data-handling Notes

Organising Raw Data

Raw data can be difficult to read. Sort values or organise them into a table before finding summaries. Ordered data makes the median, minimum, maximum and range easier to identify.

Example data:

Ordered:

Now the median is immediately visible as 11.

Mean

The arithmetic mean is:

For $6,8,9,12,15$:

The mean need not be one of the original observations.

Median with Even Number of Values

If there are an even number of observations, take the mean of the two middle values after ordering.

For:

median is:

Outliers

An outlier is a value noticeably far from most of the data. It can affect the mean and range strongly.

Consider:

The value 56 is an outlier. The median is 11, but the mean is much higher because of 56.

Dot Plot Interpretation

A dot plot preserves every data value. Look for:

• where most dots cluster;
• the most frequent values;
• empty gaps;
• possible outliers;
• overall range.

If the dots are tightly grouped, the data has less spread than a plot with dots scattered over a wider interval.

Bar-graph Scale

A bar graph can be misread if the vertical scale is ignored. Check whether marks increase by 1, 5, 10 or another amount. Also check whether the axis begins at zero.

Comparing Two Data Sets

Do not compare only the mean. Two classes can have the same mean but very different spread.

Example:

Class A: $9,10,10,10,11$

Class B: $2,6,10,14,18$

Both have mean 10, but Class A is tightly clustered while Class B is much more spread out.

15. More Data Examples

When an Outlier Appears

Now change 11 to 31:

Median remains 7, but mean becomes:

The single large value changes the mean strongly. This is why a data summary must be interpreted, not merely calculated.

Reading a Graph Carefully

When comparing bars, use the numerical scale rather than visual height alone. If the axis starts at 50 rather than 0, small numerical differences may look exaggerated.

Asking Good Data Questions

A useful statistical investigation clearly identifies:

• what is being measured;
• who or what is included;
• the units;
• how data will be recorded;
• what comparison or summary is needed.

Common Mistakes

Quick Revision

Practice Questions

Additional Practice

Chapter Summary