Statistics and Data Interpretation

Statistics questions in the FSCE 11+ exam require you to calculate averages, interpret data from charts and tables, and — critically — explain what the data means in context. The FSCE specifically tests your ability to give thoughtful, reasoned interpretations rather than just perform calculations. This lesson teaches you the mathematical skills and the communication skills you need.

Averages

An average is a single number that represents a set of data. There are three types of average you need to know, plus the range:

Mean

Mean = total of all values ÷ number of values

The mean uses every piece of data and is the most commonly used average.

Example: Test scores: 7, 8, 5, 9, 6. Mean = (7 + 8 + 5 + 9 + 6) ÷ 5 = 35 ÷ 5 = 7

Median

Median = the middle value when the data is arranged in order.

1. Put the values in order (smallest to largest).
2. If there is an odd number of values, the median is the middle one.
3. If there is an even number of values, the median is the mean of the two middle values.

Example: 3, 7, 9, 12, 15. Median = 9 (the middle value). Example: 3, 7, 9, 12. Median = (7 + 9) ÷ 2 = 8.

Mode

Mode = the value that appears most often. There can be more than one mode, or no mode at all.

Example: 2, 3, 3, 5, 7, 7, 7, 9. Mode = 7 (appears 3 times).

Range

Range = largest value - smallest value. The range measures how spread out the data is.

Example: 2, 3, 3, 5, 7, 7, 7, 9. Range = 9 - 2 = 7.

Which Average to Use?

Average	Best when...	Weakness
Mean	Data has no extreme outliers	Can be affected by very large or very small values
Median	Data has outliers or is skewed	Does not use all the data
Mode	Data is categorical (e.g. favourite colour) or you want the most common value	May not exist, or may not be representative

Reading Charts and Tables

Bar Charts

Each bar represents a category. The height (or length) of the bar shows the frequency or value. Read values carefully from the scale on the axis — check what each gridline represents.

Line Graphs

Points are connected by lines to show how a value changes over time. Look for trends: is the line going up (increasing), going down (decreasing), or staying flat (constant)?

Pie Charts

A pie chart shows proportions. The whole circle represents 100% (or the total). Each slice represents a fraction of the total.

To find the value represented by a slice:

1. Read the angle or percentage of the slice.
2. Multiply: (angle/360) x total = value, or (percentage/100) x total = value.

Interpreting Data in Context

This is where the FSCE exam differs from many other 11+ exams. You are often asked to explain what the data tells you or compare data sets. Here is how strong answers differ from weak ones:

Before/After: Weak vs Strong Interpretation

Data: A class tracked the number of books read per month. Mean in September: 2.3 books. Mean in March: 4.1 books.

Weak Answer	Strong Answer
"The March mean is higher."	"The mean number of books read increased from 2.3 in September to 4.1 in March, nearly doubling. This suggests the reading programme has been effective at encouraging pupils to read more."
"More books were read."	"On average, pupils read 1.8 more books per month in March than in September. However, the mean can be affected by outliers, so it would be useful to check whether this increase was spread across all pupils or driven by a few enthusiastic readers."

What Makes a Strong Interpretation?

1. Use specific numbers from the data.
2. Make a comparison (e.g. between two values, two time periods, or two groups).
3. Explain what it means in the real-world context.
4. Consider limitations (e.g. could outliers affect the mean?).

Misleading Statistics

Data can be presented in ways that mislead. Common tricks include:

• Broken axis: A bar chart where the y-axis does not start at zero, making differences look bigger than they are.
• Selective data: Choosing only certain data points that support a particular conclusion.
• Misleading averages: Using the mean when the median would be more appropriate (e.g. when there are outliers).
• Different sample sizes: Comparing percentages from groups of very different sizes.

Worked Examples

Worked Example 1: Calculating the Mean

Question: In five maths tests, Priya scored 78, 85, 62, 91, and 74. Calculate her mean score.

Step-by-step solution:

1. Add up all scores: 78 + 85 + 62 + 91 + 74 = 390
2. Count the number of values: 5
3. Mean = 390 ÷ 5 = 78

Answer: Priya's mean score is 78.

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Worked Example 2: Finding the Median from an Even Set

Question: The heights (in cm) of 6 plants are: 12, 8, 15, 23, 11, 18. Find the median height.

Step-by-step solution:

1. Arrange in order: 8, 11, 12, 15, 18, 23
2. There are 6 values (even number), so the median is the mean of the 3rd and 4th values.
3. 3rd value = 12, 4th value = 15.
4. Median = (12 + 15) ÷ 2 = 27 ÷ 2 = 13.5cm

Answer: The median height is 13.5cm.

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Worked Example 3: Reading a Pie Chart