The normal distribution The curve to the right was symmetrical. In this topic we will look at a particular type of symmetrical curve, the normal or bell curve. Below is an example. In normal distribution: The frequency graph is ‘bell’ shaped The mean, median and mode are all equal

The normal curve The shape varies with different means and standard deviations. The thinner the curve, the smaller the standard deviation. These 3 curves have the same mean. The fatter the curve, the larger the standard deviation. These 3 curves have different means. The curves are all the same height and width, So they have the same standard deviation.

Area under the normal curve For any normal distribution: 100% of the scores lie under the normal curve; 68% of the scores lie within 1 standard deviation of the mean, or 68% lie within ẍ ± σn; 95% of the scores lie within 2 standard deviations of the mean, or 95% lie within ẍ ± 2σn; 99·7% of the scores lie within 3 standard deviations of the mean, or 99·7 % lie within ẍ ± 3σn; 0·3% of the scores are “in the tail”. 68% 95% 99·7%

Area and probability 68% within 1SD, 34% each side We say most scores (68%) lie within 1SD of the mean. 95% within 2SD, 47·5% each side,  47·5  34 = 13·5% A score will most probably (95%) lie within 2SD’s of the mean. 34% 34% 99·7% within 3SD, 49·85% each side,  49·85  47·5 = 2·35% A score will almost probably (99 ·7%) lie within 3SD’s of the mean. 100  99·7 = 0·3% outside 3SD’s 0·15% each side 13·5 % 13·5 % 0·15 % 0·15 % 2·35 % 2·35 %

Example 1 The average mark on a history test was 71%. If the standard deviation was 6%, within what limits do 68% of the scores lie? 95% of the scores lie? 99·7% of the scores lie? You may find it useful to draw the curve and write in the values = 71 ± 6 between 65 and 77 = 71 ± 2 × 6 between 59 and 83 = 71 ± 3 × 6 between 53 and 89

Example 2 The TDK produces 240 minute tapes with a mean time of 243 minutes standard deviation of 3 minutes What percentage of tapes have between 240 and 246 minutes? What is the probability of getting a tape with less than 240 min? If they produced 15 000 tapes yesterday, how many had more than 249 min of tape? This is 1SD above and below the mean, 68% of the tapes. Less than 240 is less than 1SD below the mean, 50% - 34% = c) 249 min is 2SD above the mean 2.35% + 0.15%= 2·5 ÷ 100 × 15 000 = 16% 2·5% 375 tapes

Today’s work Exercise 8D pg 248 # 1 – 3 Exercise 8E pg #1, 3, 5, 6 – 10, 11ab, 12