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“Teach A Level Maths” Statistics 1 Estimating the Standard Deviation © Christine Crisp Estimating the Standard Deviation S1: Estimating the Standard Deviation AQA "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Estimating the Standard Deviation The formula for the standard error ( the standard deviation of the sample means ) is standard error (s.e. ) = s n where s is the population standard deviation and n is the sample size. However, we may not know the population standard deviation so we must estimate this from our sample. The obvious quantity to use is the sample standard deviation but it can be shown that this is too small so we need to make an adjustment. Estimating the Standard Deviation When we are estimating in Statistics, we talk about biased and unbiased estimators. An unbiased estimator is one that on average gives the value we are estimating. So, for example, if the value we wanted to estimate was equal to 2, and all possible samples gave us these values: 1.8, 1.9, 2, 2.1, 2.2 the statistic giving the values would be unbiased: it’s mean is 2. ( We mustn’t worry that 4 of the 5 values are wrong. That isn’t the point. We must be right on average. ) If we had 1.8, 2, 2, 2, 2.3 our estimator is biased since the average is not correct, even though more individual values are correct. We want unbiased estimators. Estimating the Standard Deviation Let’s look at the hens eggs again. We’ve met 3 different standard deviations (s.ds.) so we need to be clear which s.d. we are talking about. Population, 1st sample and Population and 1000 sample means mean of 1st sample s s n The 1st s.d. is the Population standard deviation ( the one we want to estimate ) The 2nd s.d. is the standard error or standard deviation of all sample means. It is also unknown as it depends on the unknown s. Estimating the Standard Deviation Let’s look at the hens eggs again. We’ve met 3 different standard deviations (s.ds.) so we need to be clear which s.d. we are talking about. Population, 1st sample and Population and 1000 sample means mean of 1st sample s s could be the standard deviation of this sample s n We are left with the 3rd s.d., the standard deviation, s, of our one sample. It can shown ( although we don’t need to do it ) that this is a biased estimator. However, we can tweak it to change it into an unbiased estimator. Estimating the Standard Deviation Let’s look at the hens eggs again. We’ve met 3 different standard deviations (s.ds.) so we need to be clear which s.d. we are talking about. Population, 1st sample and Population and 1000 sample means mean of 1st sample s s n n The unbiased estimator of s is s n1 where s is the standard deviation of a sample. Estimating the Standard Deviation The unbiased estimator of s 2 In your formula book you will find the unbiased estimator of s 2, the population variance, written as S2 2 ( X X ) i n1 To use this, replace the capital Xi by xi ( the sample data ) and X by x ( the sample mean ). It gives the same result as S 2 s 2 n n1 ( For standard deviation, just square root. ) However, you’ll probably be using calculator functions not a formula and your calculator gives the unbiased estimator of the population standard deviation as well as the sample standard deviation. Try the following: Estimating the Standard Deviation Enter the following data in your calculator: 1, 3, 5 Select the list of statistics and you should find the values 1 15470. . . S s 0 94280. . . s n n1 The 1st ( larger ) of these is the unbiased estimator of s. The other is the standard deviation of the sample. If you are not sure which to use, think about whether you are making estimates from a sample. If so, use the 1st ( larger ) value. n One further point: ifused n is by large, is veryThey close are to 1not . Ignore the symbols the calculator. n1 the ones we use. The biased and unbiased estimators are nearly the same. Estimating the Standard Deviation SUMMARY An unbiased estimator is one where the average of all possible values equals the quantity being estimated. To estimate the variance of a population we use the unbiased estimator, S2, where n S s n1 2 2 and s2 is the variance of a sample of size n. S2 can also be found from S 2 2 ( x x ) n1 where x represents each data item and x is the sample mean. Calculators give the values of both s, the sample standard deviation and S the unbiased estimator of population standard deviation but we must ignore the calculator notation. Estimating the Standard Deviation e.g. 1. Six people in a factory were selected and asked how long they took to get to work. The results, in minutes, were as follows: 7, 12, 13, 20, 30, 35 Calculate the mean and variance of the times in the sample and hence find unbiased estimates of the mean and variance of the times for all the workers. Solution: x Sample mean, x 19 5 n Although showing the formulae the solution, I’mm This is theI’m unbiased estimate of theinpopulation mean, 2 using the calculator functions to find each answer. x 2 2 Sample variance, s x 10 0 2 101 ( 3 s. f . ) n n 2 2 2 S s gives the ( 3 sas . f .10·0457 ) The calculator s.d. . . . so 11 0sample 121 n 1 to find the variance. I’ve written down we need to square 2, is The unbiased estimate of the variance, s 121 ( 3 s. f . ) 3 s.f. but will use the more exact calculator value. Estimating the Standard Deviation e.g. 2. The following sets of data are from samples, each from a different Normal population. Find unbiased estimates of the mean, m, and standard deviation, s, of each of the populations. (a) 17, 24, 25, 31, 42 2 (b) x x 422 , 18002 , (c) x 330, ( x x ) 2 828 , n 10 n5 Solutions: (a) Sample: x 27 8 , s 8 38 Unbiased estimates of population parameters are: m 27 8 , s 9 36 Estimating the Standard Deviation (b) 2 x x 422, 18002 , n 10 Sample mean, x x 42 2 n Unbiased estimate of mean, m is 42 2 2 x 18002 Sample variance, s 42 2 2 19 36 x 10 n n Estimate of population variance: S 2 s 2 n1 10 2 S 19 36 21 5 9 2 2 Unbiased estimate of population standard deviation, s is S 21 5 4 64 ( 3 s. f . ) Estimating the Standard Deviation (c) x 330, 2 ( x x ) 828 , n 5 Sample mean, x x 66 n Unbiased estimate of mean, m is 66 Unbiased estimate of s is S where S2 2 ( x x ) n1 207 S 207 14 4 ( 3 s. f . ) Estimating the Standard Deviation Exercise The following sets of data are from samples, each from a different Normal population. Find unbiased estimates of the mean, m, and standard deviation, s, of each of the populations. (a) 5·2, 7·9, 8·1, 9·3 2 (b) x x 678 , 52302 , (c) x 282, n 10 2 ( x x ) 1046 , n 5 Solutions: (a) Sample: x 7 625, s 1 50 Unbiased population estimates: m 7 63 ( 3 s. f . ) , s 1 73 Estimating the Standard Deviation (b) x 678, 2 x 52302 , n 10 Sample mean, x x 67 8 n Unbiased estimate of mean, m is 67 8 Sample variance, s 2 2 x n x2 52302 67 8 2 633 36 10 n S s 704 n1 2 2 Unbiased estimate of s is S 704 26 5 ( 3 s. f . ) Estimating the Standard Deviation (c) x 282, 2 ( x x ) 1046 , n 5 Sample mean, x x 56 4 n Unbiased estimate of mean, m is 56 4 Unbiased estimate of s is S where S2 2 ( x x ) n1 261 5 S 261 5 16 2 ( 3 s. f . ) Estimating the Standard Deviation The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet. Estimating the Standard Deviation The unbiased estimator of s 2 The unbiased estimator of s 2 ( the population variance ) is given by n S 2 s2 n1 In your formula book you will find this written as S2 2 ( X X ) i n1 You can use either, replacing the capital Xi by xi and X by x ( data and mean ) from your sample. However, you’ll probably be using calculator functions not a formula and your calculator the unbiased estimator of population standard deviation as well as the sample standard deviation. Estimating the Standard Deviation Enter the following data in your calculator: 1, 3, 5 Select the list of statistics and you should find the values 1 15470. . . S s 0 94280. . . s n n1 The 1st ( larger ) of these is the unbiased estimator of s. The other is the standard deviation of the sample. If you aren’t sure which to use, think about whether you are making estimates from a sample. If so, use the 1st ( larger ) value. n One further point: if n is large, is very close to 1. n1 The biased and unbiased estimators are nearly the same. SUMMARY Estimating the Standard Deviation An unbiased estimator is one where the mean of all possible values equals the quantity being estimated. To estimate the variance of a population we use the unbiased estimator, S2, where n S s n1 2 2 and s2 is the variance of a sample of size n. S2 can also be found from S 2 2 ( x x ) n1 where x represents each data item and x is the sample mean. Calculators give the values of both s, the sample standard deviation and S the unbiased estimator of population standard deviation but we must ignore the calculator notation. Estimating the Standard Deviation e.g. The following sets of data are from samples, each from a different Normal population. Find unbiased estimates of the mean, m, and standard deviation, s, of each of the populations. (a) 17, 24, 25, 31, 42 2 (b) x x 422 , 18002 , (c) x 330, ( x x ) 2 828 , n 10 n5 Solutions: (a) Using calculator functions, For the sample, x 27 8 , s 8 38 Unbiased estimates of population parameters are: m 27 8 , s 9 36 Estimating the Standard Deviation (b) 2 x x 422, 18002 , n 10 Sample mean, x x 42 2 n Unbiased estimate of mean, m is 42 2 2 x 18002 Sample variance, s 42 2 2 19 36 x 10 n n Estimate of population variance: S 2 s 2 n1 10 2 2 S 19 36 21 5 9 2 2 Unbiased estimate of population standard deviation, s is S 21 5 4 64 ( 3 s. f . ) Estimating the Standard Deviation (c) x 330, ( x x ) 2 828 , n 5 x Sample mean, x 66 n Unbiased estimate of mean, m is 66 Unbiased estimate of population standard deviation, s, is S where S2 2 ( x x ) n1 207 S 207 14 4 ( 3 s. f . )