VIEWS: 9 PAGES: 15 CATEGORY: Accounting POSTED ON: 8/29/2010
Variance and Standard Deviation (1) Spread (1) 50% Tells us the spread of the middle 50% of the data A 12 day survey of Fred and Doris’s toilet habits revealed some exciting results Spread (2) Doris's Fred visits Deviation visits to the Deviation Day to the loo from Mean loo from Mean 1 5 -3.1 28 15.4 2 11 2.9 17 4.4 3 2 -6.1 5 -7.6 4 6 -2.1 7 -5.6 5 0 -8.1 4 -8.6 6 4 -4.1 14 1.4 7 14 5.9 13 0.4 8 9 0.9 22 9.4 9 16 7.9 16 3.4 10 20 11.9 9 -3.6 11 3 -5.1 6 -6.6 12 7 -1.1 10 -2.6 Mean 8.1 Mean 12.6 Mean = sum of data / pieces of data Mean = 8.1 1.5 1 0.5 0 Fred -0.5 Doris -1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 -1.5 -2 -2.5 Mean = 12.6 Doris's Fred visits visits to the To measure spread we Day to the loo loo 1 5 28 can find the deviation 2 3 11 2 17 5 from the mean 4 5 6 0 7 4 6 4 14 The deviations may be positive 7 8 14 9 13 22 e.g. Day 10, Fred : 20 – 8.1 = 11.9 9 16 16 10 20 9 11 3 6 or negative 12 7 10 e.g. Day 10, Doris : 9 – 12.6 = -3.6 Mean 8.1 12.6 Deviations from Mean (1) Doris's Day Fred visits to the loo Deviation visits to the Deviation from Mean loo from Mean To summarise the 1 5 -3.1 28 15.4 total deviation, 2 11 2.9 17 4.4 3 2 -6.1 5 -7.6 you can’t just add 4 5 6 0 -2.1 -8.1 7 4 -5.6 -8.6 – since they may 6 7 4 14 -4.1 5.9 14 13 1.4 0.4 cancel 8 9 0.9 22 9.4 9 16 7.9 16 3.4 10 20 11.9 9 -3.6 11 3 -5.1 6 -6.6 12 7 -1.1 10 -2.6 Mean 8.1 Mean 12.6 Either: ignore the signs and add Or: square the deviations (to make positive) then add Deviations from Mean (2) Doris's Fred visits Deviation Deviation visits to the Deviation Deviation Day to the loo from Mean Squared loo from Mean Squared 1 5 -3.1 9.5 28 15.4 237.7 2 11 2.9 8.5 17 4.4 19.5 3 2 -6.1 37.0 5 -7.6 57.5 4 6 -2.1 4.3 7 -5.6 31.2 5 0 -8.1 65.3 4 -8.6 73.7 6 4 -4.1 16.7 14 1.4 2.0 7 14 5.9 35.0 13 0.4 0.2 8 9 0.9 0.8 22 9.4 88.7 9 16 7.9 62.7 16 3.4 11.7 10 20 11.9 142.0 9 -3.6 12.8 11 3 -5.1 25.8 6 -6.6 43.3 12 7 -1.1 1.2 10 -2.6 6.7 Sum of deviation squared 408.9 Sum of deviation squared 584.9 Since two sets of data may have a different number of items, you divide by the number of items in the data set … this is known as the variance Variance Variance = sum of (the deviations from mean)2 number of pieces of data For Fred, variance = 408.9 / 12 = 34.1 For Doris, variance = 584.9 / 12 = 48.7 Standard Deviation If the data were cm , then the variation would be in cm2 To keep it in the same units – we square root the variance. This is known as the Standard Deviation Standard Deviation = variance For Fred, standard deviation = variance = 34.1 = 5.8 For Doris, standard deviation = variance = 48.7 = 7.0 Activity • Page 26 of your Statistics 1 book and try … • Exercise A • Calculating Variance and Standard Deviation Notation (1) (sigma) denotes sum If data has n values - x1, x2, x3, …. , xn The sum of the data is written xi x (‘x bar’) denotes its mean x = xi (sum of numbers divided by pieces of data) n Notation (2) The sum of the squared deviations from the mean is written (xi - x)2 Hence, variance and standard deviation are : Variance = (xi - x)2 n Standard Deviation = (xi - x) 2 n Standard Deviation is denoted by s or sx Variance is usually denoted by s2 or sx2 Notation (3) We may as well get all the notation out the way Sometimes calculators or computer programs use different symbols mean x is sometimes denoted by Standard Deviation is denoted by s or sx .. but sometimes by n HAVE A LOOK AT YOUR CALCULATOR More on Standard Deviation Standard Deviation = (xi - x)2 n Look at page 27-28 of your Statistics 1 book for the proof (if I make a Slide of this, it’ll take me till midnight) Standard Deviation can more conveniently be written sx = xi 2- nx 2 or xi 2 - x 2 n n … this makes manual calculations much simpler Activity Page 28 of your Statistics 1 book and try … • Exercise C • Calculating Standard Deviation • Use your calculators • And the formulas