Range and standard deviation.

Step 1 of 5

The range

In one town the noon temperature is nearly always 20°C. In another it jumps between 5°C and 35°C. Both have the same mean, but living in them is very different. To describe data you also need a number that says how much they spread.

The simplest measure is the range: the largest value minus the smallest.

Move Scatter and watch the beads close in on the line at $5$⁠ or move away from it. Then work out the range of a short list.

Step 2 of 5

The distance from the mean

The range depends on only two values. A measure that uses all of them starts from each value’s distance from the mean, $x - \bar{x}$⁠. You cannot simply average those distances: they always add to $0$⁠, because the mean is the balance point. So square each distance, average the squares, and take the root to get back to the readings’ units:

$s = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$$s =$$\sqrt{\frac{\sum (x - \bar{x})^2}{n}}$⁠

This is the standard deviation. Its square, $s^2$⁠, is called the variance.

The board shows it literally: from each bead a stick runs to the mean line, with a square on each stick. The amber square under the plot is the average square, $s^2$⁠, and its side is $s$⁠. The blue band marks $\bar{y} \pm s$⁠.

Now work out a standard deviation by hand.

Step 3 of 5

Setting the spread

The standard deviation is measured in the same units as the readings. If every distance from the mean doubles, $s$⁠ doubles too.

Here $s = 0.5$⁠. Tune the spread.

Step 4 of 5

Range against standard deviation

The board shows $20$⁠ readings with $s = 1$⁠. Their range is $3.78$⁠, from $3.33$⁠ to $7.11$⁠.

Now the bad reading again, $0$⁠. Predict which of the two measures reacts more strongly.

Step 5 of 5

Inside the band

The standard deviation is a typical distance from the mean. Some readings are closer and some farther. How many exactly? Count on the board.