PXGZ6102 BASIC STATISTICS FOR RESEARCH IN EDUCATION Chap 3 - Measures of Variability – Standard Deviation, Variance

Measures of Variability Range Mean Deviation Variance Standard Deviation

Range Refers to the overall span of the scores Eg. 18, 34, 44, 56, 78 The range is 78 – 18 = 60

Mean Deviation Eg. Scores 1,2,3 Mean = 2 Mean deviation = |1-2 | + | 2-2 | + | 3-2 | 3 = = Mean

Mean Deviation Eg. Shoes sizes in Ali’s home: 11,12,13,14,15,16,17 the mean is 14 In Ahmad’s home: 5,8,11,14,17,20, 23 the mean is also 14 But the distribution in Ahmad’s home is greater

Calculation of Mean Deviation Ali’s Scores Mean (x - mean) |x – mean| N = 7 |x – Mean| =12 Σ |x – Mean| 12 Mean Deviation = = = 1.71 n 7

Calculate the Mean Deviation of Ahmad’s data 5,8,11,14,17,20, 23

Variance Variance of a distribution is the average of the squared deviations The formula for the variance of a population is slightly different than the formula for a sample variance

Population & Sample Variance Population Variance σ 2 = Σ (x - µ) 2 N Where σ 2 = the symbol for the population variance x = a raw score µ = the population mean N = the number of scores in the population Sample Variance Where s 2 = the symbol for sample variance = the sample mean n = the number of scores in the distribution

Why divide by (n-1) for the sample variance to estimate the population variance? Variances of samples taken from population tend to be smaller than the population variance. Dividing the sample formula with n-1 gives the correction and the actual population variance

Population Variability Population Distribution Sample Scores xx Sample A Sample B Sample C Sample scores are not as spread out as the population distribution. Thus the variance of sample tend to underestimate the variance of population. Placing n -1 in the denominator increases the value of the sample variance and provides a better estimate of the population variance

Example: Calculate the variance of the following sample scores, s 2 A) 3, 4, 6, 8, = 26 = For sample

Example: Calculate the variance of the following population scores, A) 3, 4, 6, 8, 9 XXmXm X - X m (X – X m ) = 26 = For Population

Exercise 1 1) Find the sample variance and the population variance of the following distributions A) 2, 4, 5, 7, 9 B) 22, 32, 21, 20, 19, 15, 23 C) 23, 67, 89, 112, 134, 156, 122, 45

Other forms of the Variance formula Deviation score, D = X - µ (for population) or d = X – X m (for sample) Square of deviation = (X - µ) 2 Sum of Squares of Deviation, SS = Σ (X - µ) 2 X m = sample mean

Population (N) Sample (n) Mean Deviation |X - µ| N | X – X m | n -1 Variance (Definitional Formula) σ 2 = Σ (X - µ) 2 N s 2 = Σ (X – X m ) 2 n - 1 (Deviational Formula) σ 2 = Σ D 2 N s 2 = Σ d 2 n - 1 (Sum of Squares Formula) σ 2 = SS N s 2 = SS n - 1 (Computational Formula) σ 2 = ΣX 2 – [(ΣX) 2 /N] N s 2 = ΣX 2 – [(ΣX) 2 /n] n -1 X m = Mean

If you use the Computational Formula – no need to find the Mean X X ΣX 2 ΣXΣX Is (ΣX) 2 = ΣX 2 ?

Exercise 2 Calculate the mean, variance and Standard deviation for the following distribution for sample x f σ 2 = Σ f(X - µ) 2 N s 2 = Σ f(X – X m ) 2 n - 1 X m = Mean

Standard Deviation Is the square root of the variance Is the square root of the deviational formula, sum of squares formula and computational formula of variance Example: What is the Standard Deviation of the distributions in Exercise 1?

Exercise 3 Find the Standard Deviation of the following sample distributions A) 3, 4, 5, 7, 8, 9 B) 12, 23, 34, 56, 13, 24

Exercise 4 What are the Standard Deviations of the distributions in Exercise 2 were those of the population?

Exercise 5 1) Calculate the range, mean deviation, variance and standard deviation of this sample of scores 13, 18, 3, 23, 6, 12, 34 2) The distribution of Maths marks for class 4A is as follows: 34, 45, 23, 47, 12, 67, 89. Calculate the range, mean deviation, variance and standard deviation using both the definitional formula and the computational formula

Standard Deviation for Grouped Data – Example for Population Class Interval f Midpoint fm d Deviation fD D 2 fD 2 (m d ) D = m d - X m 60 – – – σ = ΣfD 2 – ΣfD 2 N N N = 10 X m = 710 = Σ fD = 0Σ fD 2 = 190 σ = = √19 = 4.36 X m = Grouped mean

Exercise 6 Find the mean, variance and standard deviation of the following grouped data (population): Class Interval Frequency – – 19 3

Exercise 7 Find the mean, variance and standard deviation of the following distribution using the grouped calculation method 78, 74, 80, 65, 63, 74, 67, 58, 74, 65, 65, 63, 74, 86, 80, 74, 67, 50, 78, 89

Exercise 8 Find the mean, variance and standard deviation of the following distribution

To display variability – Box-and- whisker plot (refer to Chapter 1)

End of Chapter 3 – Measures of Variability