Five-Number Summary, Box Plots & Data Analysis
Master the five-number summary, box-and-whisker plots, and interpreting data spread.
Statistics
Table of Contents
Statistics: Central Tendency, Spread & Box Plots#
Statistics turns raw data into meaningful information. In Grade 10, you learn to summarise data using measures of centre (where the data clusters) and measures of spread (how far it stretches).
Measures of Central Tendency#
| Measure | What it finds | How to calculate |
|---|---|---|
| Mean ($\bar{x}$) | The “fair share” average | $\bar{x} = \frac{\text{sum of all values}}{n}$ |
| Median | The middle value | Sort data, find the middle position |
| Mode | The most frequent value | Count which value appears most often |
Finding the Median#
- Sort the data from smallest to largest
- If $n$ is odd: median = the middle value (position $\frac{n+1}{2}$)
- If $n$ is even: median = average of the two middle values
The Five-Number Summary#
| Value | What it is |
|---|---|
| Minimum | Smallest value |
| $Q_1$ (Lower Quartile) | Median of the bottom half (25th percentile) |
| $Q_2$ (Median) | Middle value (50th percentile) |
| $Q_3$ (Upper Quartile) | Median of the top half (75th percentile) |
| Maximum | Largest value |
Measures of Spread#
| Measure | Formula | What it tells you |
|---|---|---|
| Range | Max $-$ Min | Total spread of the data |
| IQR | $Q_3 - Q_1$ | Spread of the middle 50% |
💡 The IQR is more useful than the range because it ignores extreme values (outliers).
Box-and-Whisker Plots#
A box plot is a visual summary of the five-number summary:
- Box: Stretches from $Q_1$ to $Q_3$ (the middle 50%)
- Line inside box: The median
- Whiskers: Extend to the minimum and maximum
Reading a Box Plot#
| Feature | Interpretation |
|---|---|
| Median centred in box | Data is symmetric |
| Median closer to $Q_1$ | Data is positively skewed (tail to the right) |
| Median closer to $Q_3$ | Data is negatively skewed (tail to the left) |
| Long whisker | Extreme values in that direction |
Grouped Data#
When data is given in class intervals (e.g., 40–50, 50–60, …), you cannot find the exact mean or median. Instead:
- Use the midpoint of each class to estimate the mean
- Use the ogive (cumulative frequency curve) to estimate the median and quartiles
Deep Dive#
- Five-Number Summary, Box Plots & Data Analysis — full worked examples for calculating the five-number summary, drawing box plots, and interpreting data
🚨 Common Mistakes#
- Not sorting data first: You MUST sort data before finding the median and quartiles.
- Including the median in quartile calculations: When finding $Q_1$ and $Q_3$, split the data into two halves. If $n$ is odd, exclude the median from both halves.
- Confusing mean and median: The mean is affected by extreme values; the median is not. If asked “which is the better measure?”, consider outliers.
- Box plot scale: Draw the number line to scale — the box plot must be proportional.
🔗 Related Grade 10 topics:
- Probability — data analysis connects to probability
📌 Where this leads in Grade 11: Statistics: Standard Deviation — measuring spread numerically with variance and $\sigma$