# Data, Summary Statistics and Bivariate Data

> Mean, median, quartiles, box plots, standard deviation, distribution shape, correlation and lines of best fit.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/data-and-statistics/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The two coffee carts
Two coffee carts park outside the station. Over ten mornings, both sell 80 coffees on average. Cart A sells between 76 and 84 every day. Cart B sells 40 one day and 120 the next.

Same average, very different businesses. Cart A can order the right amount of milk every time. Cart B keeps running out or wasting stock. An average alone does not tell the full story. You also need to know how **spread out** the data is.

**In short:** Summary statistics describe a data set with a centre (mean, median, mode) and a spread (range, interquartile range, standard deviation). For two variables, a scatter plot and the correlation $r$ show how strongly they are linked, and a line of best fit makes predictions.

## Starting point: Years 9 and 10

### Measures of centre

- **Mean** $\bar{x}$: add the values and divide by how many there are.
- **Median:** the middle value when the data is in order. For an even count, average the two middle values.
- **Mode:** the most common value.

The median is not affected much by an extreme value, so it is better for skewed data such as house prices.

### Measures of spread

- **Range** = highest $-$ lowest.
- **Quartiles:** $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half.
- **Interquartile range** $IQR = Q_3 - Q_1$, the spread of the middle 50%.

### Worked example: Five-number summary
Data: 3, 5, 7, 8, 9, 12, 13, 15, 21.

Minimum 3, $Q_1 = 6$, median 9, $Q_3 = 14$, maximum 21. $IQR = 8$.

### Box plots

*Diagram: Box plot with minimum 3, Q1 6, median 9, Q3 14 and maximum 21*
*Minimum, Q1, median, Q3 and maximum. The box holds the middle half of the data.*

**Outliers** are values below $Q_1 - 1.5 \times IQR$ or above $Q_3 + 1.5 \times IQR$. Here the fences are $-6$ and 26, so there are no outliers.

### Standard deviation

The **standard deviation** $\sigma$ measures the typical distance of the data from the mean. Cart A has a small standard deviation. Cart B has a large one. Calculators find it in statistics mode.

### Calculator tip: Statistics mode
In statistics mode (`STAT`, then `1-VAR`), type the data into the list. The results menu gives $\bar{x}$ (the mean), $\sigma x$ (the population standard deviation) and $sx$ (the sample standard deviation). Use $\sigma x$ unless the question says the data is a sample. For a frequency table, switch on the frequency column in the setup menu.

For bivariate data, choose `A+BX` mode and enter both lists. The menu gives $r$ (the correlation coefficient) and the least-squares line $y = a + bx$.

### Shape

- **Symmetric:** mean $\approx$ median.
- **Positively skewed:** a long tail to the right, mean pulled above the median.
- **Negatively skewed:** a long tail to the left.

### Key idea: Centre and spread together
Always describe data with a centre **and** a spread: mean with standard deviation, or median with IQR.

### Watch out: Order the data first
The median and quartiles only work on data sorted from smallest to largest.

**Mind map**

```text
Statistics
  Centre
    Mean
    Median
    Mode
  Spread
    Range
    IQR
    Standard deviation
  Displays
    Box plot
    Histogram
    Outlier fences
  Shape
    Symmetric
    Skewed
  Two variables
    Scatter plot
    Correlation r
    Line of best fit
```

### Year 11: Grouped data and percentiles
For a frequency table, the mean is

$$\bar{x} = \frac{\sum f x}{\sum f}$$

For grouped data use the class centre as $x$.

A **cumulative frequency** graph (ogive) shows how many values are at or below each point. Read across from 50% for the median, from 25% and 75% for the quartiles. The $k$th **percentile** is the value below which $k\%$ of the data falls.

**Changing every score.** Adding a number $k$ to every score adds $k$ to the mean, median and quartiles, but leaves the range, IQR and standard deviation unchanged. Multiplying every score by $k$ multiplies the mean, median, range, IQR and standard deviation by $k$, and the variance (the standard deviation squared) by $k^2$.

**Combining groups.** Use totals, not an average of averages. 20 students with mean 60 and 30 with mean 70 have a total of $1200 + 2100 = 3300$, so the combined mean is $\dfrac{3300}{50} = 66$.

**A removed value.** Ten numbers with mean 12 total 120. If nine of them have mean 11 (total 99), the removed number was $120 - 99 = 21$. Adding a new value equal to the mean leaves the mean the same but makes the standard deviation a little smaller.

**Comparing with z-scores.** To compare results from different tests, measure each in standard deviations from its mean: $z = \dfrac{x - \bar{x}}{\sigma}$. A score of 80 where the mean is 70 and $\sigma = 5$ has $z = 2$, which beats 85 in a test with mean 75 and $\sigma = 10$ ($z = 1$).

### Year 12: Bivariate data
**Bivariate** data has two variables for each item, like hours studied and test score. Plot them on a **scatter plot**.

**Pearson's correlation coefficient** $r$ measures how close the points are to a straight line:
- $r$ near 1: strong positive. $r$ near $-1$: strong negative. $r$ near 0: no linear relationship.

The **least-squares regression line** $\hat{y} = a + bx$ is the line of best fit, found on a calculator. Use it to predict.

- **Interpolation** (inside the data range) is reliable.
- **Extrapolation** (outside the range) is risky.

**Correlation is not causation.** Ice cream sales and shark sightings rise together, but both are caused by hot weather.

**Facts about the least-squares line.** It always passes through the point of means $(\bar{x}, \bar{y})$. If the line is $\hat{y} = 1.5x + c$ with $\bar{x} = 10$ and $\bar{y} = 22$, then $22 = 15 + c$ and $c = 7$.

**Interpreting the gradient.** The gradient is the change in $y$ for each extra unit of $x$. In $P = 30T - 150$ for profit and temperature, each extra degree adds about 30 dollars of profit (within the range of the data).

**Coefficient of determination.** $r^2$ is the fraction of the variation in $y$ explained by the line. If $r^2 = 0.81$ and the line slopes down, then $r = -0.9$.

## Quick summary

- Mean, median and mode measure the centre. Range, IQR and standard deviation measure spread.
- Five-number summary: minimum, $Q_1$, median, $Q_3$, maximum. Draw it as a box plot.
- Outliers lie beyond $Q_1 - 1.5 \times IQR$ or $Q_3 + 1.5 \times IQR$.
- Skewed data pulls the mean towards the tail. Use the median there.
- For two variables: scatter plot, correlation $r$ and a least-squares line.

## Quiz

Try these questions from the chapter quiz.

1. Find the mean of 4, 7, 9, 10, 15.
   - A. 7
   - B. 9
   - C. 10
   - D. 45
   - Answer: B. 9. $4 + 7 + 9 + 10 + 15 = 45$, and $45 \div 5 = 9$.

2. Find the median of 3, 8, 1, 6, 10, 4.
   - A. 6
   - B. 5
   - C. 4.5
   - D. 7
   - Answer: B. 5. Sorted: 1, 3, 4, 6, 8, 10. Average of 4 and 6.

3. What is the mode of 2, 5, 5, 6, 7, 7, 7, 9?
   - A. 5
   - B. 7
   - C. 6
   - D. 9
   - Answer: B. 7. 7 appears most often.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/data-and-statistics/#quiz