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Slide 1 - Normal Distributions and z-scores Hello 2014 – Further maths
Slide 2 - Normal Distributions and z-scores Hello 2014 – Further maths The normal distribution
Slide 3 - Normal Distributions and z-scores Hello 2014 – Further maths 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:
Slide 4 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side)
Slide 5 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side)
Slide 6 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side)
Slide 7 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 50% of values lie either side of the mean
Slide 8 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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
Slide 9 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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
Slide 10 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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 20 134 154 114 94 174 134 134 194 74 Diagram 1 Diagram 3 Diagram 2 Diagram 4 134
Slide 11 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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 20 134 154 114 94 174 134 134 194 74 Diagram 1 Diagram 3 Diagram 2 Diagram 4 134 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)
Slide 12 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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 20 134 154 114 94 174 134 134 194 74 Diagram 1 Diagram 3 Diagram 2 Diagram 4 134 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) 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
Slide 13 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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 20 134 154 114 94 174 134 134 194 74 Diagram 1 Diagram 3 Diagram 2 Diagram 4 134 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) 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 5 170 155 160 165 185 180 175
Slide 14 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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 20 134 154 114 94 174 134 134 194 74 Diagram 1 Diagram 3 Diagram 2 Diagram 4 134 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) 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 5 170 155 160 165 185 180 175 Example 2 – solutions a) 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% We already found that 16% are likely to have heights above 170cm. 16% of 5000 = 0.16 x 5000 = 800 So 800 women are expected to have heights above 170cm
Slide 15 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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 20 134 154 114 94 174 134 134 194 74 Diagram 1 Diagram 3 Diagram 2 Diagram 4 134 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) 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 5 170 155 160 165 185 180 175 Example 2 – solutions a) 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% We already found that 16% are likely to have heights above 170cm. 16% of 5000 = 0.16 x 5000 = 800 So 800 women are expected to have heights above 170cm 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
Slide 16 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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 20 134 154 114 94 174 134 134 194 74 Diagram 1 Diagram 3 Diagram 2 Diagram 4 134 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) 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 5 170 155 160 165 185 180 175 Example 2 – solutions a) 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% We already found that 16% are likely to have heights above 170cm. 16% of 5000 = 0.16 x 5000 = 800 So 800 women are expected to have heights above 170cm 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:
Slide 17 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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 20 134 154 114 94 174 134 134 194 74 Diagram 1 Diagram 3 Diagram 2 Diagram 4 134 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) 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 5 170 155 160 165 185 180 175 Example 2 – solutions a) 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% We already found that 16% are likely to have heights above 170cm. 16% of 5000 = 0.16 x 5000 = 800 So 800 women are expected to have heights above 170cm 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: 53 68 75
Slide 18 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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 20 134 154 114 94 174 134 134 194 74 Diagram 1 Diagram 3 Diagram 2 Diagram 4 134 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) 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 5 170 155 160 165 185 180 175 Example 2 – solutions a) 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% We already found that 16% are likely to have heights above 170cm. 16% of 5000 = 0.16 x 5000 = 800 So 800 women are expected to have heights above 170cm 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: 53 68 75 Example 1 – solution a) so a score value of 53 is two standard deviations below the mean b) So a value of 68 is 0.5 standard deviations above the mean c) So a value of 75 is 1.67 standard deviations above the mean
Slide 19 - Normal Distributions and z-scores Hello 2014 – Further maths 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: 68% of values lie within 1 standard deviation of the mean (with 16% of values either side) 95% of values lie within 2 standard deviations of the mean (with 2.5% of values either side) 99.7% of values lie within 3 standard deviations of the mean (with 0.15% of values either side) 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 20 134 154 114 94 174 134 134 194 74 Diagram 1 Diagram 3 Diagram 2 Diagram 4 134 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) 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 5 170 155 160 165 185 180 175 Example 2 – solutions a) 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% We already found that 16% are likely to have heights above 170cm. 16% of 5000 = 0.16 x 5000 = 800 So 800 women are expected to have heights above 170cm 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: 53 68 75 Example 1 – solution a) so a score value of 53 is two standard deviations below the mean b) So a value of 68 is 0.5 standard deviations above the mean c) So a value of 75 is 1.67 standard deviations above the mean Holiday homework Holiday homework must be completed neatly and accurately and submitted first lesson back in 2014. It is a work requirement and must be completed. You will be tested on chapters 1-3 in the first week back. Your holiday homework is all questions from the chapter reviews from chapters 1, 2 and 3