Summary statistics: Centre and spread for VCE General Mathematics Unit 3 Data analysis

What this dot point is asking

Once a numerical distribution has been displayed, it is summarised with numbers. VCAA wants you to calculate and interpret measures of centre (the mean $\bar{x}$ and the median) and spread (the sample standard deviation $s$, the interquartile range and the range), to know which measures suit which shapes of distribution, and to use them to compare distributions in terms of centre and spread. The study design lists "the sample mean and standard deviation and their use in comparing data distributions in terms of centre and spread" explicitly, alongside the five-number summary (which has its own boxplots page).

The means and standard deviations are calculated with the CAS in the examinations, but you must know what they mean, when to use them, and how outliers affect them. VCAA reports repeatedly show marks lost for rounding when not asked to, reading the wrong subgroup, and using the wrong statistic in a comparison.

The answer

Measures of centre

Mean. The sum of the values divided by the number of values:

The mean uses every value, so it is sensitive to extreme values: one very large value pulls it up.

Median. The middle value of the ordered data. For $n$ values, it is at position $\frac{n+1}{2}$: the middle value when $n$ is odd, and the average of the two middle values when $n$ is even. The median only depends on the order, so it is resistant to outliers.

Mode. The most frequent value (or category). Useful mainly for categorical data or discrete data with many repeats.

Measures of spread

Range $=$ maximum $-$ minimum. Quick, but determined entirely by the two most extreme values.

Interquartile range $\text{IQR} = Q_3 - Q_1$: the spread of the middle $50\%$ of the data. Resistant to outliers. (VCAA's method for quartiles: split the ordered data into a lower and upper half, leaving out the median when $n$ is odd, and take the median of each half.)

Sample standard deviation:

It measures the typical distance of the values from the mean. Every deviation contributes (squared), so an extreme value has a large effect. The CAS reports it as $s_x$; do not use $\sigma_x$, which divides by $n$.

What the standard deviation means

For 4, 6, 7, 9, 14, the mean is 8 and the deviations are $-4, -2, -1, 1, 6$ (they always add to zero). Squaring and adding gives 58, so

A typical value is about 3.8 units from the mean. A standard deviation of 0 means every value is the same; a larger standard deviation means the values are more spread out. For bell-shaped data the standard deviation also has a precise meaning through the 68-95-99.7% rule (see the normal distribution page): about $68\%$ of values lie within one standard deviation of the mean.

Set A (6, 7, 8, 9, 10) has $s \approx 1.58$; set B (4, 6, 7, 9, 14) has $s \approx 3.81$. Same mean, very different spread.

How outliers and skew affect the measures

The hourly pay of 10 staff is 42, 45, 47, 48, 50, 51, 53, 55, 58 and 160 dollars.

One extreme value moves the mean by about $11 and multiplies the standard deviation by seven, but barely changes the median or IQR. That is why skewed data and data with outliers are summarised with the median and IQR.

In a skewed distribution the mean is pulled towards the tail: positive skew usually means mean $>$ median; negative skew usually means mean $<$ median. Comparing the mean and median is a quick check of skewness.

Working backwards from a mean

Because $\text{mean} \times n = \text{total}$, many questions can be answered by working with totals. If the mean of six attempts must be exactly 8, the total must be $6 \times 8 = 48$; if the first five total 39, the sixth must be 9 (2023 Examination 1 Question 3). Similarly, adding a new value changes the mean by $\frac{\text{new value} - \text{old mean}}{n + 1}$.

Changing every value

• Adding a constant $k$ to every value adds $k$ to the mean, median and quartiles but leaves the standard deviation, IQR and range unchanged (the data just shifts).
• Multiplying every value by $k$ (for example converting units) multiplies every measure of centre and spread by $k$ (for positive $k$).

Comparing distributions

When two or more groups are compared, the report should cover centre and spread, with values:

"The median apartment price ($775 000) is lower than the median house price ($1 136 500). Apartment prices have a larger standard deviation ($346 466 compared with $300 911), so they are more variable."

Use the pair of statistics the question gives or asks for. If a question says "using the information in Table 2", use only those statistics, as the 2025 examiners stressed.

Using the CAS

Enter the data as a list, then use one-variable statistics. Read $\bar{x}$ (mean), $s_x$ (sample standard deviation), $n$, minimum, $Q_1$, median, $Q_3$ and maximum. Check the value of $n$ to make sure every value was entered, since a single keying error changes the answers (the 2025 report specifically mentioned careful data entry).