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12 FURTHER MATHEMATICS The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Mean and the Standard Deviation The Normal Distribution The Normal Distribution Standard deviation is a measure of the spread of a data distribution about the average (mean) For normal distributions, we can determine the % of values that lie within 1, 2 or 3 standard deviations of the mean: The Normal Distribution 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) The Normal Distribution 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) The Normal Distribution 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) The Normal Distribution 50% of values lie either side of the mean Summary Diagram The diagram below can be used in all cases. It shows the % of values that lie in each section of the normal distribution curve Example 1 We always start by drawing diagrams. Each diagram below has been scaled with a mean of 134 and standard deviation of 20 Diagram 1 Diagram 3 114 134 154 Diagram 2 94 74 134 Diagram 4 134 174 134 194 The Normal Distribution Example 1 – solutions a) About 68% of executives have blood pressure between 114 and 154 (diagram 1) b) About 95% of executives have blood pressure between 94 and 174 (diagram 2) c) About 99.7% of executives have blood pressure between 74 and 194 (diagram 3) d) About 16% of executives have blood pressures above 154 (diagram 1) e) About 2.5% of executives have blood pressures below 94 (diagram 2) f) About 0.15% of executives have blood pressures below 74 (diagram 3) g) About 50% of executives have blood pressures above 134 (diagram 4) The Normal Distribution Example 2 For this question we could again use the four separate diagrams that were used in example 1, but instead we can use the summary diagram (see next page). Again a scale is included for a mean of 170 and standard deviation of 5 The Normal Distribution The Normal Distribution Example 2 – solutions a) vi. Between 155cm and 185cm: 2.35% + 13.5% + 34% + 34% + 13.5% + 2.35% = 99.7% More than 175cm: 13.5% + 2.35% + 0.15% = 16% More than 170cm: 34% + 13.5% + 2.35% + 0.15% = 50% Less than 160cm: 2.35% + 0.15% = 2.5% Less than 165cm: 13.5% + 2.35% + 0.15% = 16% Between 160cm and 180cm: 13.5% + 34% + 34% + 13.5% = 95% b) We already found that 16% are likely to have heights above 170cm. 16% of 5000 = 0.16 x 5000 = 800 i. ii. iii. iv. v. So 800 women are expected to have heights above 170cm The Normal Distribution The advantage of the summary chart is for a question like this (not symmetrical): What % of these women have heights between 165cm and 180cm? Answer: 34% + 34% + 13.5% = 81.5% 81.5% of women have heights between 165cm and 180cm Z-scores We can also determine how many standard deviations above or below the mean a value lies by standardising the data. The standardised scores are called z-scores. To calculate z-score: Example 1 A set of data has a mean of 65 and a standard deviation of 6. Standardise the following scores: a) b) c) 53 68 75 WORK TO BE COMPLETED Chapter 3

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