STANDARD DEVIATION Calculating and understanding standard deviation as

STANDARD DEVIATION Calculating and understanding standard deviation as a measure of spread.

Consider these exam results in 8 subjects for 2 different pupils: Alex : - 45, 56, 76, 65, 63, 71, 90, 46 Barbara : - 50, 52, 95, 92, 56, 51, 60, 56 Which student did better overall? Calculate the mean and range to find out.

Results The mean for both Alex and Barbara is 64. The range for both pupils is 45. How can we decide which pupil got the more consistent scores? The standard deviation tells us how far on average the numbers are away from the mean. In Statistics we use the symbol (x bar) to represent the mean.

Calculating standard deviation Complete the table for Alex’s scores. = 64 x x - x (x - x)2 45 45 - 64 = -19 (-19)2 = 361 56 76 65 63 71 90 46

x x - x (x - x)2 45 45 - 64 = -19 (-19)2 = 361 56 -8 64 76 12 144 65 1 1 63 -1 1 71 7 49 90 26 676 46 -18 324 This column is how far the numbers are from the mean. We square the numbers so that they are all positive. Remember a negative number squared equals a positive.

x x - x (x - x)2 45 45 - 64 = -19 (-19)2 = 361 56 -8 64 76 12 144 65 1 1 63 -1 1 71 7 49 90 26 676 46 -18 324 Σ(x - x)2 = 1620 We now need to add up this column to get the total of all the distances from the mean (squared). Remember Σ means sum of.

Using the formula One of the ways the standard deviation formula can be written is: Calculated in the table σ = Small sigma - Symbol used in Maths for standard deviation. Number of items in the list Square root as we squared all the differences in the table

Standard deviation for Alex Substituting the values into the formula gives: σ = = 14. 23 (2 dp) So Alex’s marks on average are 14. 23 away from his mean of 64.

Standard Deviation for Barbara x x - x (x - x)2 50 52 95 92 56 51 60 56 Σ(x - x)2 = Repeat the steps to find the s. d. for Barbara.

Standard Deviation for Barbara x x - x (x - x)2 50 -14 196 52 -12 144 95 31 961 92 28 784 56 -8 64 51 -13 169 60 -4 16 56 -8 64 Σ(x - x)2 = 2398 s. d = √(2398/8) = 17. 31 (2 dp)

Comparing Standard Deviations Alex has a standard deviation of 14. 23 and Barbara’s is 17. 31. As Barbara’s is higher this tells us that on average her scores are further from the mean of 64 and are therefore less consistent. She did very well in 2 subjects which increased the mean but her results have a wider spread than Alex. A smaller standard deviation means more consistent results.

Alternative Formula Sometimes it is better to use the alternative formula for standard deviation. Both formula are provided on the formula sheet. This formula is quicker to calculate and is useful for when you are provided with summary data. For example a question my tell you the value of Σx 2 and ask you to calculate the standard deviation.

Using the other formula • x x 2 45 2025 56 3136 76 5776 65 4225 63 3969 71 5041 90 8100 46 2116 Σx 2 = 34388 This part of the formula is the mean

Standard Deviation from Grouped Frequency Tables •

Example The heights of 100 Y 12 pupils are recorded in the table below. Calculate the mean and standard deviation. Height (h) Frequency (f) 140<h≤ 150 6 150<h≤ 160 26 160<h≤ 170 38 170<h≤ 180 18 180<h≤ 190 12 Σf = Midpoint (x) fx Σfx = x 2 fx 2 Σfx 2 =

Example The heights of 100 Y 12 pupils are recorded in the table below. Calculate the mean and standard deviation. Height (h) Frequency (f) Midpoint (x) 140<h≤ 150 6 145 150<h≤ 160 26 155 160<h≤ 170 38 165 170<h≤ 180 18 175 180<h≤ 190 12 185 Σf = 100 fx Σfx = x 2 fx 2 Σfx 2 =

Example • Height (h) Frequency (f) Midpoint (x) fx 140<h≤ 150 6 145 870 150<h≤ 160 26 155 4030 160<h≤ 170 38 165 6270 170<h≤ 180 18 175 3150 180<h≤ 190 12 185 2220 Σf = 100 Σfx =16540 x 2 fx 2 Σfx 2 =

Example • Height (h) Frequency (f) Midpoint (x) fx 140<h≤ 150 6 145 870 150<h≤ 160 26 155 4030 160<h≤ 170 38 165 6270 170<h≤ 180 18 175 3150 180<h≤ 190 12 185 2220 Σf = 100 16540 100 Σfx =16540 x 2 fx 2 Σfx 2 =

Example The heights of 100 Y 12 pupils are recorded in the table below. Calculate the mean and standard deviation. Height (h) Frequency (f) Midpoint (x) fx x 2 140<h≤ 150 6 145 870 21025 150<h≤ 160 26 155 4030 24025 160<h≤ 170 38 165 6270 27225 170<h≤ 180 18 175 3150 30625 180<h≤ 190 12 185 2220 34225 Σf = 100 Σfx =16540 fx 2 Σfx 2 =

Example The heights of 100 Y 12 pupils are recorded in the table below. Calculate the mean and standard deviation. Height (h) Frequency (f) Midpoint (x) fx x 2 fx 2 140<h≤ 150 6 145 870 21025 126150 150<h≤ 160 26 155 4030 24025 624650 160<h≤ 170 38 165 6270 27225 1034550 170<h≤ 180 18 175 3150 30625 551250 180<h≤ 190 12 185 2220 34225 410700 Σf = 100 σ = Σfx =16540 Σfx 2 =2747300

Example The heights of 100 Y 12 pupils are recorded in the table below. Calculate the mean and standard deviation. Height (h) Frequency (f) Midpoint (x) fx x 2 fx 2 140<h≤ 150 6 145 870 21025 126150 150<h≤ 160 26 155 4030 24025 624650 160<h≤ 170 38 165 6270 27225 1034550 170<h≤ 180 18 175 3150 30625 551250 180<h≤ 190 12 185 2220 34225 410700 Σf = 100 Σfx =16540 Σfx 2 =2747300 σ = = 10. 76 cm (2 d. p. )

Using Summary Data Exam questions on standard deviation can be asked where part of the information in the formula is given. This is called summary data. Example A researcher captures at random a sample of 200 garden worms and measures their length in mm. She calculates: Σx = 12800 and Σx 2 = 819364 Calculate the mean and standard deviation of the lengths.

Using Summary Data • 12800 200

Using Summary Data Exam questions on standard deviation can be asked where part of the information in the formula is given. This is called summary data. Example A researcher captures at random a sample of 200 garden worms and measures their length in mm. She calculates: Σx = 12800 and Σx 2 = 819364 Calculate the mean and standard deviation of the lengths. σ

Using the Statistics Function on a Calculator for mean and Standard Deviation To calculate standard deviation for a list of data on a scientific calculator follow these steps: Turn on Statistics mode MODE 2 1 Choose SD Type in your x values. Type each value and press equals. SHIFT AC 1 Press then to get statistics options. 5 2 3 Choose S-VAR then for s. d or for mean.

Calculator for grouped data For grouped data you will need to have the frequency turned on in Stats mode. To do this press SHIFT MODE 3 then for frequency on. 1 Follow the steps on the previous slide to calculate the s. d. or mean. Remember x is the midpoint. Type in all the x values first then use the arrow key to get to the frequency column.

Changes to the data If the data you are using changes, this can affect the mean, median, range and standard deviation. Example 1 Mr Burns has marked his classes English essays and he has calculated the following: Mean score = 64 Median Score = 65 deviation = 7 Range = 31 Standard He then realises that he has forgotten to add on extra marks for spelling, punctuation and grammar. As he is in a hurry he decides to give all the pupils in the class 5 extra marks. What effect will this have on: a) the mean? c) the range? b) the median? d) the standard deviation?

Solution a. If he adds 5 extra marks onto each score the mean will also increase by 5 marks. For example, if there were 20 pupils in the class the original total of the scores would be: 64 x 20 = 1280 He then adds 5 marks an extra 20 times giving a total of: 1280 + (20 x 5) = 1380 The new mean is 1380 ÷ 20 = 69 a. The median score will also increase by 5 marks. If each score goes up by 5 the middle score also increases by 5. The new median = 70

Solution c) The range scores in the class will remain the same. For example, the range was 31 so the lowest and highest scores could have been 50 and 81. If he increases both scores by 5 marks the new range will be 86 – 55 = 31. d) As the standard deviation is also a measure of spread it will not be affected by adding 5 marks to each score. As the standard deviation measures the average distance from the mean and the mean score is also 5 marks higher, the standard deviation will still be 7 marks.

Example 2 Mrs Simpson has marked her students Maths tests out of 50. She has calculated the following: Mean = 35, Median = 34, Range = 14, Standard Deviation = 5 She then decides to convert them to percentages by multiplying each score by 2. What effect will this have on: a) the mean? c) the range? b) the median? d) the standard deviation?

Solution a. The mean score will double. If all the scores are doubled and then divided by the number of pupils (which remains the same), the mean score must double also. The new mean = 70 a. The new median score will also double. The middle score has doubled along with all the other scores. The new median = 68

Solution c) The new range has also doubled. As the range was 14 the highest and lowest scores could have been 42 and 28. If the scores are doubled the new range will be 84 – 56 = 28 d) The standard deviation will also double. If we consider one possible score, for example 40, it was 5 marks away from the mean. If all scores are doubled, the score will now be 80 and the mean is 70. The distance from the mean score is now 10 which is double the previous amount. This will be true for all scores so the average distance from the mean has doubled. The new standard deviation is 10.

Rules for changes to data If a constant value is added or subtracted to each value in a data set, the mean and median will increase or decrease by the constant value but the standard deviation and range will not change. If each value in a data set is multiplied or divided by a constant amount, the mean, median, standard deviation and range will all be multiplied or divided by the same amount.

Rules given algebraically •