0.1 Data summaries and variation

1. What does a summary describe?

A dataset contains observations of a variable, such as journey time, height or the number of books borrowed. Record the unit and the number of observations, $n$. Before calculating, ask what one observation represents. A summary is useful only when its meaning remains attached to the numbers.

A measure of location describes where the data are centred. A measure of spread describes how much the observations differ. Two groups can have the same mean but very different consistency. Neither a mean alone nor a spread alone gives the whole description.

2. Mean and median

The notation $\sum x$ means the sum of all observed values. The arithmetic mean, written $\bar{x}$, shares that total equally between the $n$ observations. It uses every value, so a single unusually large value can move it substantially.

To find the median, put the observations in ascending order. For odd $n$, select the value at position $(n+1)/2$. For even $n$, average the values at positions $n/2$ and $n/2+1$. The median identifies the middle of the ordered data and is less affected by an extreme value.

3. Variance and standard deviation

A deviation is an observation minus the mean. Positive and negative deviations cancel when added, so their sum cannot measure spread. Squaring makes every contribution nonnegative and gives larger deviations more influence.

For the values $2,4,4,6$, the mean is $4$. Their squared deviations are $4,0,0,4$, so $v=8/4=2$ and the standard deviation is $\sqrt{2}\approx1.41$. A variance of zero means every observation has the same value.

4. Frequencies and summary totals

A frequency $f$ tells you how many times a value occurs. Multiply each value by its frequency when calculating the total. For the square total, use $fx^2$: square the observation, then multiply by its frequency.

5. Combining and coding data

To combine groups, add their totals and their sizes. A group of four observations with mean $10$ contributes $40$ to the total. A group of six with mean $15$ contributes $90$. Their combined mean is $(40+90)/(4+6)=13$. Simply averaging the two means would give equal weight to unequal groups.

Coding replaces inconvenient values with simpler ones. If $x=a+by$, the shift $a$ changes location and the scale factor $b$ changes spread. A negative scale reverses the order of values, but variance and standard deviation remain nonnegative.

For example, if $y=x-50$, $n=5$, $\sum y=10$ and $\sum y^2=30$, then $\bar{y}=2$ and $v_y=6-4=2$. Adding $50$ back gives $\bar{x}=52$, while $v_x=2$. A shift leaves all distances from the mean unchanged.

6. Description and estimation use different denominators

This topic describes the observations you have, using division by $n$. Later, a random sample will help estimate the variance of a larger population. For independent observations from the same population with finite variance, the usual unbiased variance estimate uses $n-1$, where $n>1$.

The observed mean $\bar{x}$ is a number calculated from data. In later topics, $\bar{X}$ will denote the random sample mean before the observations are collected. Keeping this distinction clear helps connect descriptive calculations to inference.

Practice

Try each question before opening its worked solution.