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Standard Deviation

Standard Deviation. Symbols used. x (x bar) – the mean: The sum of all the items ÷ number of items. x – the variable: it may be height, weight, time etc.  (sigma) – the sum of:  x means add up all the items in the column x. Symbols used. n – the number of items:.

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Standard Deviation

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  1. Standard Deviation

  2. Symbols used x (x bar) – the mean: The sum of all the items ÷ number of items x – the variable: it may be height, weight, time etc.  (sigma)– the sum of: x means add up all the items in the column x.

  3. Symbols used n – the number of items: Example x : 7 , 8 , 8 , 10 , 12 x : 7 + 8 + 8 + 10 + 12 = 45 n = 5

  4. Standard Deviation is a measure of how values are spread out about the mean Higher Standard Deviation Low Standard Deviation High Standard Deviation

  5. Standard Deviation for a Sample of Data In real life situations it is normal to work with a sample of data ( survey / questionnaire ). We will use this version because it works for all types of questions. There are two formulae we can use to calculate the sample deviation. Both are given in the formulae list. s = standard deviation  = The sum of x = sample mean n = number in sample

  6. Standard Deviation Eight athletes have heart rates 70, 72, 73, 74, 75, 76, 76 and 76. Heart Rate x x2 4900 70 72 5184 73 5329 5476 74 5625 75 76 5776 5776 76 5776 76 43842

  7. Example 1b : Eight office staff train as athletes. Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM 6400 6561 6889 8100 8836 9216 9216 10000 x = 720 65218

  8. Who are fitter the athletes or staff. Compare means What does the deviation tell us. Staff data is more spread out. Athletes are fitter Athletes Staff Mean = 74 bpm Mean = 90 bpm Std Dev = 2·20 bpm Std Dev = 7·73 bpm

  9. Calculate the mean and Standard Deviation for each sample

  10. 1. A sample of fifteen matchboxes was chosen from an assembly line. The number of matches in each box was recorded. The results are shown below. • 65 63 66 62 59 61 62 62 • (a) Calculate, correct to 3 significant figures, the mean and standard deviation of this sample. • (b) Boxes which contain less than one standard deviation of matches below the mean are to be discarded and refilled. • How many boxes in this sample would be discarded?

  11. A group of students took part in an end of year assessment. The marks out of fifty for a sample of these students are listed below. 38 21 44 48 35 32 45 45 39 31 • (a) Calculate, correct to 3 significant figures, the mean and standard deviation of this sample. • (b) A “highly commended” certificate is to be given to any student who has a mark that is more than one standard deviation above the mean. • How many students in this sample would receive this certificate?

  12. Standard Deviation Using the formula when given ∑ x and ∑ x2

  13. Using the formula : Sometimes the values of x and x2 are given and the above formula needs to be used to calculate s, the Standard Deviation.

  14. Use the formula to work out the standard deviation in each of the examples below. • In a race the 8 finalists gave the following time data ∑ x = 360·2∑ x2 = 16223·2 • Calculate the mean and standard deviation giving your answers correct to 1 decimal place.  s = 0·9 Mean = 360·2 ÷ 8 = 45·0

  15. In a class of 10 pupils, the heights data gave the following • results • ∑ x = 618 ∑ x2 = 42475 • Calculate the mean and standard deviation giving your answers correct to 1 decimal place. 3. In a class of 12 pupils the marks in a test gave the data ∑ x = 192 ∑ x2 = 3678 Calculate the mean and standard deviation giving your answers correct to 1 decimal place. • In a survey to see how many books were carried to school by • 10 pupils the data obtained gave the results • ∑ x = 48 ∑ x2 = 269 • Calculate the mean and standard deviation giving your answers correct to 1 decimal place.

  16. In a 400m race the 7 finalists gave the following time • (seconds) data∑ times = 384 ∑ times2 = 21082 • Calculate the mean and standard deviation giving your answers correct to 1 decimal place. • The heights ‘h’ in cm of 10 plants gave the following • results ∑ h = 66 ∑ h2 = 456.16 • Calculate the mean and standard deviation of the plant heights giving your answers correct to 1 decimal place. • In a class of 6 pupils doing a sixth year maths exam the results data showed that • ∑ marks = 404 ∑ marks2 = 29686 • Calculate the mean and standard deviation of the marks giving your answers correct to the nearest whole number.

  17. 8. In a survey to see how much television is watched by pupils in a single night, the data produced by 12 pupils gave the following results ∑ hours = 46.9 ∑ hours2 = 207.31 Calculate the mean and standard deviation giving your answers correct to 2 decimal places. • The heights ‘h’ in metres of 8 pupils gave the following • results ∑ heights = 14.76 ∑ heights2 = 30.144 • Calculate the mean and standard deviation of the heights giving your answers correct to 1 decimal place. • The weights ‘w’ in kg of the same 8 pupils gave the • following results ∑ weights = 2259 ∑ weights2 = 641047 • Calculate the mean and standard deviation of the weights giving your answers correct to 1 decimal place.

  18. Answers to Using the Formula

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