Frequencies, measures of centre and measures of spread

Frequencies

A frequency table lists each value once with the number of times it occurs. That number is the absolute frequency $f_i$. Divide it by the total number of observations $n$ to get the relative frequency $\frac{f_i}{n}$, a share that can be written as a fraction, a decimal or a percent. Absolute frequencies add up to $n$; relative frequencies add up to 1. Checking that sum is the quickest way to catch a misread table.

The cumulative frequency of a value counts the observations at or below it. It is what you need to find a median from a table.

Joint and conditional frequencies

When each observation is classified in two ways, a two-way table holds the counts. In the table below, 50 customers of a shop are classified by day and by how they paid.

	Card	Cash	Total
Weekday	18	12	30
Weekend	14	6	20
Total	32	18	50

• A joint frequency is an inner cell: 14 customers paid by card at the weekend, a joint relative frequency of $\frac{14}{50} = 0.28$.
• A marginal frequency is a row or column total: 32 paid by card.
• A conditional frequency is a cell divided by its row or column total, restricted to one group: among weekend customers, $\frac{14}{20} = 70\%$ paid by card; among card payers, $\frac{14}{32} = 43.75\%$ came at the weekend.

These are the same ideas as conditional probability, with counts in place of probabilities; the only question is always which total sits in the denominator.

Measures of centre

• Mean: the sum of the values divided by how many there are. From a frequency table, $\bar{x} = \frac{\sum f_i x_i}{n}$: multiply each value by its frequency, add, and divide by the total frequency $n$, not by the number of rows.
• Median: the middle value once the data are in order. With $n$ observations, it is the value in position $\frac{n+1}{2}$; when $n$ is even, that falls between two positions and the median is the mean of the two middle values. From a frequency table, use cumulative frequencies to see which value occupies the middle positions.
• Mode: the most frequent value. A data set can have more than one.

The mean uses every value, so a single extreme value can drag it a long way; the median does not move when the largest value grows. For data with a long tail on one side, mean and median can differ a lot.

Measures of spread

• Range: largest minus smallest. Easy, but decided by two values only.
• Variance: the mean of the squared deviations from the mean. In this lesson we use the population formula, which divides by $n$:

$$\sigma^2 = \frac{1}{n}\sum (x_i - \bar{x})^2.$$

Some books divide by $n - 1$ for a sample; if a question does not say, look at the answer options, but here and in every example below the divisor is $n$.

• Standard deviation: the square root of the variance, $\sigma = \sqrt{\sigma^2}$. It is in the same units as the data, which is why it is the measure people quote.

An equivalent formula is often quicker by hand: the variance equals the mean of the squares minus the square of the mean,

$$\sigma^2 = \frac{1}{n}\sum x_i^2 - \bar{x}^2.$$

Use one formula and check with the other.

Shifting and scaling the data

If every value is transformed as $y = a x + b$:

Measure	New value
mean, median, mode	$a \cdot (\text{old}) + b$
range, standard deviation	$\lvert a \rvert \cdot (\text{old})$
variance	$a^2 \cdot (\text{old})$

Adding a constant moves every value by the same amount, so the centre moves and the spread does not. Multiplying stretches the distances between values, so the spread is multiplied too. These rules turn many statistics questions into one line of work.

Working without a calculator

Choose the route with the smallest numbers. Subtract a convenient value from every observation before averaging, then add it back: the mean of 197, 199, 204 is $200 + \frac{-3 - 1 + 4}{3} = 200$. Deviations from the mean are usually small whole numbers; squaring those is faster and safer than squaring the raw data.