The Normal Distribution and Z-Scores

0:00Heights in a large group average 170 centimetres. How unusual is 186 centimetres?

0:08Heights like these roughly follow a normal distribution: a symmetric bell curve.

0:15The peak is at the mean, 170 centimetres.

0:19The standard deviation, or SD, measures how spread out the heights are. Here it is 8 centimetres.

0:28About 68 percent of heights are within 1 standard deviation of the mean, between 162 and 178.

0:38About 95 percent are within 2 standard deviations, between 154 and 186.

0:46And about 99.7 percent are within 3, between 146 and 194.

0:54A z-score counts how many standard deviations a value is above or below the mean.

1:00For 186, subtract the mean: 186 minus 170 is 16. Then divide by the SD: 16 divided by 8 is 2.

1:13Check: 170 plus 2 times 8 is 186.

1:20The other 5 percent is outside 2 SDs, half at each end, so about 2.5 percent are taller than 186.

1:29So 186 centimetres is taller than about 97.5 percent of people.

1:36Now use z-scores to compare: is 78 on test A better than 85 on test B?

1:44Test A has mean 70 and SD 4. 78 minus 70 is 8, and 8 divided by 4 is 2.

1:54Test B has mean 75 and SD 10. 85 minus 75 is 10, and 10 divided by 10 is 1.

2:0478 is 2 SDs above its mean, but 85 is only 1, so 78 is the better result.

2:13A common mistake is comparing raw marks: 85 is higher than 78, but the tests have different means and SDs.

2:23Recap: a z-score counts SDs from the mean, so compare z-scores, not raw marks.