VARIABILITY: Range Variance Standard Deviation

Transcription

1 VARIABILITY: Range Variance Standard Deviation

2 Measures of Variability Describe the extent to which scores in a distribution differ from each other.

3 Distance Between the Locations of Scores in Three Distributions

4 Three Variations of the Normal Curve

5 The Range, Variance, and Standard Deviation

6 The Range The range indicates the distance between the two most extreme scores in a distribution Range = highest score lowest score

7 Variance and Standard Deviation The variance and standard deviation are two measures of variability that indicate how much the scores are spread out around the mean We use the mean as our reference point since it is at the center of the distribution

8 The Sample Variance and the Sample Standard Deviation

9 Sample Variance The sample variance is the average of the squared deviations of scores around the sample mean Definitional formula X X S ( ) X N

10 Sample Variance Computational formula S X X (X N N )

11 Sample Standard Deviation The sample standard deviation is the square root of the sample variance Definitional formula ( X X ) S X N

12 Sample Standard Deviation Computational formula S X X (X ) N N

13 The Standard Deviation The standard deviation indicates the average deviation from the mean, the consistency in the scores, and how far scores are spread out around the mean

14 Normal Distribution and the Standard Deviation

15 Normal Distribution and the Standard Deviation Approximately 34% of the scores in a perfect normal distribution are between the mean and the score that is one standard deviation from the mean.

16 Standard Deviation and Range For any roughly normal distribution, the standard deviation should equal about one-sixth of the range.

17 The Population Variance and the Population Standard Deviation

18 Population Variance The population variance is the true or actual variance of the population of scores. ( X ) X N

19 Population Standard Deviation The population standard deviation is the true or actual standard deviation of the population of scores. X ( X ) N

20 The Estimated Population Variance and The Estimated Population Standard Deviation

21 Estimating the Population Variance and Standard Deviation The sample variance ( S X is a biased estimator of the population variance. ) The sample standard deviation biased estimator of the population standard deviation. ( S X ) is a

22 Estimated Population Variance By dividing the numerator of the sample variance by N - 1, we have an unbiased estimator of the population variance. Definitional formula s X ( X N X 1 )

23 Estimated Population Variance Computational formula s X X N ( X N 1 )

24 Estimated Population Standard Deviation By dividing the numerator of the sample standard deviation by N - 1, we have an unbiased estimator of the population standard deviation. Definitional formula s X ( X N X 1 )

25 Estimated Population Standard Deviation Computational formula s X X N ( X ) N 1

26 Unbiased Estimators is an unbiased estimator of s X is an unbiased estimator of s X The quantity N - 1 is called the degrees of freedom

27 S X S X s X Uses of,,, and s X Use the sample variance sample standard deviation S X S X and the to describe the variability of a sample. s X Use the estimated population variance and the estimated population s X standard deviation for inferential purposes when you need to estimate the variability in the population.

28 Organizational Chart of Descriptive and Inferential Measures of Variability

29 Applying to Research 5 item list 10-item list 15-item list X S x X S x X S x The standard deviation in each condition tells me about: 1. on average the scores differ from each other (i.e. consistency of scores and behavior). the strength of overall relationship 3. amount of error we have in prediction (Rather, the variance is the average error when using the mean to predict scores)

30 Proportion of Variance Accounted For When describing a relationship, we evaluate its scientific usefulness: How important is it? What does it buy me? Using a relationship helps us predict more accurately but more accurate compared to what?

31 Proportion of Variance Accounted For Compare our average prediction error when using the relationship to the average prediction error without using the relationship S x 15 X 5 item 10-item 15-item X 3 6 Average Error = 10 X X 9

32 Proportion of Variance Accounted For The proportion of variance accounted for by a relationship is: the proportion of error in our predictions when we use the overall mean to predict scores that is eliminated when we use the relationship with another variable to predict scores i.e. the improvement that results from using a relationship to predict scores, compared to not using that relationship

33 Example 1 Using the following data set, find The range, The sample variance and standard deviation, The estimated population variance and standard deviation

34 Example1 The range is the largest value minus the smallest value

35 Example 1 S X X (X N N ) S X 3406 (46)

36 Example 1 S X X (X N N ) S X

37 Example 1 s X X N ( X N 1 ) s X 3406 (46)

38 Example 1 s X X ( X N N 1 ) s X

39 Example For the following sample data, compute the range, variance and standard deviation range= 11-6=5 Variance=.60 Standard Deviation= 1.61

40 Example 3 For the data set below, calculate the mean, deviation, sum of squares, variance and standard deviation by creating a table

41 Example 3 Solution score mean deviation sum of squares variance standard deviation 11 14,60-3,60 1, ,60 -,60 6, ,60-1,60, ,60-1,60, ,60 0,40 0, ,60 0,40 0, ,60 1,40 1, ,60 1,40 1, ,60,40 5, ,60 3,40 11,56 46,40 4,64,15

42 Example 4 For the data set below, calculate the mean, deviation, sum of squares, variance and standard deviation by creating a table

43 Example 4 Solution score mean deviation sum of squares variance standard deviation 1,50-1,50,5 1,50-1,50,5,50-0,50 0,5,50-0,50 0,5,50-0,50 0,5 3,50 0,50 0,5 3,50 0,50 0,5 3,50 0,50 0,5 4,50 1,50,5 4,50 1,50,5 10,50 1,05 1,0

44 Example 5 For the data set below, calculate the mean, deviation, sum of squares, variance and standard deviation by creating a table

45 Example 5 Solution score mean deviation sum of squares variance standard deviation 1 10,50-9,50 90,5 10,50-8,50 7,5 3 10,50-7,50 56,5 4 10,50-6,50 4,5 5 10,50-5,50 30,5 1 10,50 1,50, ,50,50 6, ,50 4,50 0,5 0 10,50 9,50 90, ,50 19,50 380,5 790,50 79,05 8,89

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