1 of 6 AS-Level Maths: Statstcs 1 for Edecel S1. Calculatng means and standard devatons Ths con ndcates the slde contans actvtes created n Flash. These actvtes are not edtable. For more detaled nstructons, see the Gettng Started presentaton.
Contents Means Calculatng means Calculatng standard devatons Codng of 6
Mean 3 of 6 The mean s the most wdely used average n statstcs. It s found by addng up all the values n the data and dvdng by how many values there are. Notaton: If the data values are mean s Ths s the mean symbol 1,, 3,..., n... n n n 1 3, then the Note: The mean takes nto account every pece of data, so t s affected by outlers n the data. The medan s preferred over the mean f the data contans outlers or s skewed. Ths symbol means the total of all the values
Mean 4 of 6 If data are presented n a frequency table: Value Frequency f 1 1 f n f n then the mean s f f... f 1 1 n n f f f
5 of 6 Mean Eample: The table shows the results of a survey nto household sze. Fnd the mean sze. Household sze, Frequency, f 1 0 8 3 5 4 19 5 16 6 6 f 0 56 75 76 80 36 TOTAL 114 343 To fnd the mean, we add a 3 rd column to the table. Mean = 343 114 = 3.01
Contents Standard devaton Calculatng means Calculatng standard devatons Codng 6 of 6
Standard devaton 7 of 6 There are three commonly used measures of spread (or dsperson) the range, the nter-quartle range and the standard devaton. The standard devaton s wdely used n statstcs to measure spread. It s based on all the values n the data, so t s senstve to the presence of outlers n the data. The varance s related to the standard devaton: varance = (standard devaton) The followng formulae can be used to fnd the varance and s.d. varance ( ) ( ) s.d. n n
Standard devaton 8 of 6 Eample: The md-day temperatures (n C) recorded for one week n June were: 1, 3, 4, 19, 19, 0, 1 Frst we fnd the mean: 1 3... 1 147 1 C 7 7 1 0 0 3 4 4 3 9 19-4 19-4 0-1 1 1 0 0 Total: ( ) ( ) varance n So varance = 7 = 3.143 So, s.d. = 1.77 C (3 s.f.)
Standard devaton 9 of 6 There s an alternatve formula whch s usually a more convenent way to fnd the varance: ( ) varance n But, ( ) ( ) n n n n Therefore, varance n and s.d. n
Standard devaton 10 of 6 Eample (contnued): Lookng agan at the temperature data for June: 1, 3, 4, 19, 19, 0, 1 We know that 147 1 C 7 Also, 1 3... 1 = 3109 So, 3109 varance 1 3. 143 n 7 s.d. 1. 77 C Note: Essentally the standard devaton s a measure of how close the values are to the mean value.
Calculatng standard devaton from a table 11 of 6 When the data s presented n a frequency table, the formula for fndng the standard devaton needs to be adjusted slghtly: s.d. f f Eample: A class of 0 students were asked how many tmes they eercse n a normal week. Fnd the mean and the standard devaton. Number of tmes eercse taken Frequency 0 5 1 3 5 3 4 4 5 1
Calculatng standard devaton from a table 1 of 6 No. of tmes eercse taken, Frequency, f f f 0 5 1 3 5 3 4 4 5 1 0 0 3 3 10 0 1 36 8 3 5 5 TOTAL: 0 38 116 The table can be etended to help fnd the mean and the s.d. 38. 0 19 f 116 s.d. 1. 9 14. 8 f 0
13 of 6 Calculatng standard devaton from a table If data s presented n a grouped frequency table, t s only possble to estmate the mean and the standard devaton. Ths s because the eact data values are not known. An estmate s obtaned by usng the md-pont of an nterval to represent each of the values n that nterval. Eample: The table shows the annual mleage for the employees of an nsurance company. Estmate the mean and standard devaton. Annual mleage, Frequency 0 < 5000 6 5000 < 10,000 17 10,000 < 15,000 14 15,000 < 0,000 5 0,000 < 30,000 3
Calculatng standard devaton from a table 14 of 6 Mleage Frequency, f Md-pont, f f 0 5000 6 500 15000 37,500,000 5000 10,000 17 7500 17,500 956,50,000 10,000 15,000 14 1,500 175,000,187,500,000 15,000 0,000 5 17,500 87,500 1,531,50,000 0,000 30,000 3 5,000 75,000 1,875,000,000 TOTAL 45 480,000 6,587,500,000 480,000 10,667 mles 45 6,587,500,000 s.d. 10,667 45 5711 mles
Notes about standard devaton 15 of 6 Here are some notes to consder about standard devaton. In most dstrbutons, about 67% of the data wll le wthn 1 standard devaton of the mean, whlst nearly all the data values wll le wthn standard devatons of the mean. Values that le more than standard devatons from the mean are sometmes classed as outlers any such values should be treated carefully. Standard devaton s measured n the same unts as the orgnal data. Varance s measured n the same unts squared. Most calculators have a bult-n functon whch wll fnd the standard devaton for you. Learn how to use ths faclty on your calculator.
Eamnaton-style queston Eamnaton-style queston: The ages of the people n a cnema queue one Monday afternoon are shown n the stem-and-leaf dagram: 16 of 6 3 means 3 years old 3 6 3 1 6 6 4 1 5 6 9 5 0 4 7 6 1 a) Eplan why the dagram suggests that the mean and standard devaton can be sensbly used as measures of locaton and spread respectvely. b) Calculate the mean and the standard devaton of the ages. c) The mean and the standard devaton of the ages of the people n the queue on Monday evenng were 9 and 6. respectvely. Compare the ages of the people queung at the cnema n the afternoon wth those n the evenng.
Eamnaton-style queston a) The mean and the standard devaton are approprate, as the dstrbuton of ages s roughly symmetrcal and there are no outlers. 597 14 b) 597 so, 4. 6486 46. 17 of 6 3 means 3 years old 3 6 3 1 6 6 4 1 5 6 9 5 0 4 7 6 1 7,131 7131 so, s.d. 4. 6486 10. 9 14 c) The cnemagoers n the evenng had a smaller mean age, meanng that they were, on average, younger than those n the afternoon. The standard devaton for the ages n the evenng was also smaller, suggestng that the evenng audence were closer together n age.
18 of 6 Combnng sets of data Sometmes n eamnaton questons you are asked to pool two sets of data together. Eample: S male and fve female students st an A-level eamnaton. The mean marks were 5% and 57% for the males and females respectvely. The standard devatons were 14 and 18 respectvely. Fnd the combned mean and the standard devaton for the marks of all 11 students.
Combnng sets of data 19 of 6 Let Let y,..., 1 6,..., y 1 5 be the marks for the 6 male students. be the marks of the 5 female students. To fnd the overall mean, we frst need to fnd the total marks for all 11 students. As 5 65 31 As y 57 y 557 85 Therefore y 31 85 597 So the combned mean s: 597 54. 77.... 3 % 11 54
Combnng sets of data 0 of 6 To fnd the overall standard devaton, we need to fnd the total of the marks squared for all 11 students. Notce that the formula s.d. n rearranges to gve As s.d. 14 As s.d. y 18 Therefore, n( s.d. ) 6( 14 5 ) 17,400 y 5( 18 57 ) 17,865 y 35,65 So the combned s.d. s: 35,65 54. 7 1 61. % 11 (to 3 s.f.)
Contents Codng Calculatng means Calculatng standard devatons Codng 1 of 6
Codng of 6 Codng s a technque that can smplfy the numercal effort requred n fndng a mean or standard devaton. Enter some data below, and see how t changes when you add or multply by dfferent numbers.
Codng 3 of 6 Addng So, f a number b s added to each pece of data, the mean value s also ncreased by b. The standard devaton s unchanged. Multplyng If each pece of data s multpled by a, the mean value s multpled by a. The standard devaton s also multpled by a. More formally, f s.d. y a b y a b y as.d. then:
Codng 4 of 6 Eample: Fnd the mean and the standard devaton of the values n the table. Use the transformaton below to help you. 1 y 5 10 Frequency 50 3 60 5 70 7 80 4 90 1 y 0 1 3 4 Usng the gven transformaton, add a y column to the table.
Codng 5 of 6 y Frequency, f 0 3 1 5 7 3 4 4 1 y f y f 0 0 5 5 14 8 1 36 4 16 Total 0 35 85 To fnd the mean: y 35. 0 1 75 To fnd the s.d.: f y 85 s.d. y 1. 75 1. 09 f 0
Codng 6 of 6 You have now found the mean and standard devaton of y. To fnd them for the values, you must reverse the codng. 1 We can rearrange: y 5 10 to get: 10y50 Therefore the mean of s: 10y 50 101. 75 50 67. 5 And the standard devaton of s: 10 1.09 = 10.9 Note how the codng helped to smplfy the calculatons by makng the numbers smaller.