1. TOPIC 13 Standard Deviation

2. The STANDARD DEVIATION is a measure of dispersion and it allows us to assess how spread out a set of data is: 1. STANDARD DEVIATION FOR A SET OF NUMBERS The formula used to calculate the STANDARD DEVIATION of a SET OF NUMBERS is: Standard Deviation (SD) = √ ∑x 2 - ( ∑x ) 2 n (n) Or SD = √ ∑x 2 - x 2 n where, x = individual data values n = number of data values x = mean

3. Standard Deviation For a Set of Numbers Example 1 Calculate the standard deviation of this set of numbers: 179, 86, 137, 140, 86, 104, 125 Answer 1 SD = √ ∑x 2 - ( ∑x) 2 n (n) = √111643 – (857) 2 7 (7) = √15949 – 122.429 2 = √15949 – 14988.7551 = √960. 245 = 30.99 xx2x2 17932041 867396 13718769 14019600 867396 10410816 12515625 ∑x = 857∑x 2 = 111643

4. Standard Deviation For a Set of Numbers Another important measure in statistics is the VARIANCE. VARIANCE = (STANDARD DEVIATION) 2 Therefore, for a SET OF NUMBERS: Variance = ∑x 2 - ( ∑x ) 2 n (n) So for Example 1, variance = 960.245 Note:Adding the same number to (or subtracting the same number from) all data values has no effect on the SD. Multiplying (or dividing) all the data values by the same number means the SD is also multiplied (or divided) by this number.

5. Standard Deviation For a Frequency Distribution 2. STANDARD DEVIATION FOR FREQUENCY DISTRIBUTION The formula used to calculate the STANDARD DEVIATION of a FREQUENCY DISTRIBUTION is: Standard Deviation (SD) = √ ∑fx 2 - ( ∑fx ) 2 n (n) Or SD = √ ∑fx 2 - x 2 n where, x = data values f = frequency n = total frequency x = mean

6. Standard Deviation For a Frequency Distribution Example 2 Find the standard deviation of the following distribution of the number of children per family. Answer 2 Children (x)Frequency (f)x2x2 fx 2 fx 05000 1161 22248844 3897224 45168020 53257515 6136 6 n = 60∑fx 2 = 367∑fx = 125

7. Standard Deviation For a Frequency Distribution Answer 2 SD = √ ∑fx 2 - ( ∑fx) 2 n (n) = √367 – (125) 2 60(60) = √6.117 – 2.083 2 = √6.117 – 4.339 = √1.778 = 1.33

8. Standard Deviation For a Grouped Frequency Distribution 3. STANDARD DEVIATION FOR GROUPED FREQUENCY DISTRIBUTION The formula used to ESTIMATE the STANDARD DEVIATION of a GROUPED FREQUENCY DISTRIBUTION is also: Standard Deviation (SD) = √ ∑fx 2 - ( ∑fx ) 2 n (n) Or SD = √ ∑fx 2 - x 2 n where, x = midpoint of group f = frequency of group n = total frequency x = mean

9. Standard Deviation For a Grouped Frequency Distribution Example 3 Find an estimate for the standard deviation of the following distribution. Answer 3 Age (years)Frequency (f)Midpoint (x)x2x2 fx 2 fx 0-48243216 5-91174953977 10-1413121441872156 15-1919172895491323 20-247224843388154 25-29227729145854 n = 60∑fx 2 = 12780 ∑fx = 780

10. Standard Deviation For a Grouped Frequency Distribution Answer 3 SD = √ ∑fx 2 - ( ∑fx) 2 n (n) = √12780 – (780) 2 60 (60) = √213 – 13 2 = √213 – 169 = √44 = 6.63 Variance = ∑fx 2 - ( ∑fx ) 2 = 44 n (n)