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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
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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.
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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%
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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 %
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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
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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 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% %= 2·5 ÷ 100 × = 16% 2·5% 375 tapes
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Today’s work Exercise 8D pg 248 # 1 – 3 Exercise 8E pg
#1, 3, 5, 6 – 10, 11ab, 12
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