The 68-95-99.7% Rule and z-scores

Normal distribution• The shape of histograms sometimes approximate what is known as a “bell-shape.”

Normal distribution• By joining the columns of the histogram together,

a bell-shaped curve is formed.• When we get data that forms a bell-shape, we call

it a normal distribution.• Two important things to remember about normal

distributions:– The mean of the data, , is at the centre of the bell-

curve.– Most of the data is gathered close to the mean. The

further we go from the mean the less data there is.

x

The 68-95-99.7% rule• Recall that standard deviation, s, is a measure of the spread of the data.• In a normal distribution, 68% of data have values within 1 standard deviation of the mean.

The 68-95-99.7% rule95% of the data have values within 2 standard deviations of the mean.

The 68-95-99.7% rule99.7% of the data have values within 3 standard deviations of the mean.

The 68-95-99.7% rule• Using the 68-95-99.7% rule, we can work out the percentage of data in each section of the bell curve.

Questions• Examples on board 1I, Q. 2, 3, 4, 10, 13• Questions: 5, 7, 9, 11, 14

z-scores• The z-score indicates the position of a certain score in a set of data relative to the mean. • It enables us to compare scores in different distributions.• A z-score of zero means that the score is equal to the mean. A negative z-score means that the score is below the mean, and a positive z-score is a score above the mean.

z-scores• Calculate z-scores using the formula

where x is the score, is the mean, and s is the standard deviation.• A z-score of -1 would mean that the score is one standard deviation below the mean. A score that is one standard deviation above the mean would have a z-score of +1.

x xz s x

Questions• Examples: 15, 20, 23• Questions: 16, 19, 22, 25