Step 5: Describing Data with Numbers

Measures of centre: mean, median and mode

Picture a town of 1,000 households with a typical income of about 50,000 dollars. A billionaire registers as a resident. The mean income jumps to over a million dollars, yet no family is a cent richer, and the household in the middle still earns about 50,000. The mean was dragged by one extreme value; the median did not move.

Here \( \Sigma \) (capital sigma) means “add up”, \( \bar{x} \) (“x-bar”) is the sample mean and \( n \) is the number of values. For a whole population the mean is written \( \mu \) (“mu”) and the size \( N \); the arithmetic is the same. For example, five students who studied 4, 8, 6, 5 and 12 hours have a mean of \( 35 \div 5 = 7 \) hours.

The median is the middle value once the data is in order. If \( n \) is odd, it is the single middle value; if \( n \) is even, it is the average of the two middle values. For 4, 6, 9, 10 the median is \( (6 + 9) \div 2 = 7.5 \). The mode is the most frequent value, and it is the only measure of centre that works for categorical data, such as the most common eye colour.

On symmetric data the three measures nearly coincide. On skewed data they separate in a predictable way: the mean is pulled towards the long tail, the median stays near the bulk of the data, and the mode sits at the peak. A quick trick:

• mean > median: probably right-skewed (incomes, house prices);
• mean < median: probably left-skewed;
• mean ≈ median: roughly symmetric.

Measures of position: percentiles and quartiles

If you are “in the 90th percentile” for an exam, about 90% of students scored at or below you.

A simple hand method: find the median, then \( Q_1 \) is the median of the lower half and \( Q_3 \) is the median of the upper half. The five-number summary (minimum, \( Q_1 \), median, \( Q_3 \), maximum) is the skeleton behind a boxplot.

Measures of spread: range, IQR and standard deviation

Two classes could both average 65%, but in one everyone scored 60 to 70 while in the other scores ran from 20 to 100. A measure of spread captures that difference.

• The range = maximum − minimum. It uses only two values, so one outlier can blow it up.
• The interquartile range (IQR) = \( Q_3 – Q_1 \), the spread of the middle 50%. It ignores the extremes, so it pairs naturally with the median.

The most important measure is the standard deviation. Start with each value’s deviation from the mean, \( x – \bar{x} \). If you simply averaged these they would cancel to zero. So you square each deviation first (making it positive), average the squares to get the variance, then take the square root to get back to the original units.

Why \( n – 1 \)? A sample sits slightly closer to its own mean than to the population mean, so dividing by \( n \) would underestimate the spread.

The coefficient of variation, \( \text{CV} = (s \div \bar{x}) \times 100\% \), expresses spread as a percentage of the mean, so you can compare things measured in different units. Coffee sales averaging 200 cups with SD 30 have CV \( 30 \div 200 = 15\% \); revenue averaging 4,000 dollars with SD 480 has CV \( 480 \div 4000 = 12\% \), so coffee sales are relatively more variable.

Z-scores and the empirical rule

A student scores 82 in maths (class mean 70, SD 8) and 78 in English (class mean 60, SD 12). Which result is better? A z-score puts both on the same scale.

For data that is roughly bell-shaped (symmetric with one peak), the empirical rule (the 68–95–99.7 rule) says about 68% of values lie within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD. IQ scores have mean 100 and SD 15, so about 95% of people score between 70 and 130, and only about 2.5% score above 130 (half of the 5% outside).

Outliers and choosing what to report

To decide whether a value is “far from the rest”, use the IQR fences:

Anything beyond a fence is an outlier. For the summary above, \( 1.5 \times 9 = 13.5 \), so the fences are \( 7.5 – 13.5 = -6 \) and \( 16.5 + 13.5 = 30 \). The maximum, 21, is inside; a value of 35 would be flagged.

The key idea when choosing summaries is resistance: whether a measure is thrown off by outliers.

Keep the pairs together: mean with SD, median with IQR. And when you change units, adding a constant shifts the mean but not the SD, while multiplying by a constant scales both.

Practise: 6 questions

Try each one on paper first, then check the worked answer.

Check yourself

The short version

• Mean, median, mode: the mean is pulled by outliers, the median resists them, the mode works for categories.
• Quartiles give the five-number summary; IQR \( = Q_3 – Q_1 \) is the resistant spread.
• Standard deviation: square the deviations, divide by \( n – 1 \) for a sample, take the square root.
• Z-scores compare across scales; the 68–95–99.7 rule applies to bell-shaped data; outliers lie beyond the 1.5 × IQR fences.
• Report mean and SD for symmetric data, median and IQR for skewed data. The full chapter in Statistics Made Simple has over 50 practice questions with worked answers.