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VARIABILITY

VARIABILITY. Measure of Variability. A measure of variability is a summary of the spread of performance. Suppose that 2 students took 10 quizzes in a Preparatory School. Measure of Variability. The pass-point of the preparatory school is 70.

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VARIABILITY

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  1. VARIABILITY

  2. Measure of Variability • A measure of variability is a summary of the spread of performance. • Suppose that 2 students took 10 quizzes in a Preparatory School.

  3. Measure of Variability • The pass-point of the preparatory school is 70. • What do you think, which student would be more likely to pass 70 in the final exam? • Compare two distributions

  4. Measure of Variability

  5. Measure of Variability • As you can see, the distributions of the scores for two prep students differ, even if their means are identical. • It seems that the first student will not succeed in the final exam, since his/her score was never better than the cut point, 70 point. • Maybe, English is too hard for him/her. • However, the second student pass 70 point in 5 quizzes (50% of the tests). • So, it is more likely for him/her to pass 70 point in the final exam. • Today, many online shopping sites provide information about their costumers’ ratings for the products. Which would be more informative for you, a measure of tendency for the ratings, or a measure of variability?

  6. The Measures of Variability • Range • The Inter-quartile Range • Variance • Standard Deviation

  7. Range • The range is the difference between the lowest and highest values in a dataset. • The range of the first students scores is 61 – 45 = 16 • The range of the second students scores is 87 – 12 = 75 • The range is based solely on the two most extreme values within the dataset • exceptionally high or low scores (outliers) will result in a range that is not typical of the variability within the dataset. • In order to reduce the problems caused by outliers in a dataset, the inter-quartile range could be calculated instead of the range

  8. The Inter-quartile Range (IQR) • The inter-quartile range is a measure that indicates the extent to which the central 50% of values within the dataset are dispersed. • To calculate the inter-quartile range, we need to subtract the lower quartile from the upper quartile • Q3 – Q1 = P75 – P25 • The IQR for first student’s scores is • Q1 = 48 • Q3 = 55.5 • IQR = Q3 - Q1 = 55.5 – 48 = 7.5

  9. Variance • Variance is a deviation score. It summarize the amount of deviation from mean • Note: If you are interested in inferential statistics, then you should divide the sum of squared deviation by n-1, rather than n!

  10. Standard Deviation • Variance is a squared value of measurement. For that reason, it is not appropriate for descriptive statistics. • Stud1 • Variance = 24.45 • SD = 4.94

  11. Properties of Range • Range is easy to compute and it is good for a fast scan of the data • Range is based on two extreme scores. So, it does not say anything about the rest of the scores • Range has little use beyond the descriptive level

  12. Properties of IQR • Semi-Interquantile Range is quite similar to median. So, it is not sensitive to the exact values of the scores, but their ranks • Therefore, it is more resistant to the presence of a few extreme scores (compared to variance)

  13. Properties of Standart Deviation • The standart deviation, like the mean, is responsive to the exact position/value of every score in the distribution • Similarly, it is more sensitive to the extreme values. • Standart deviation is resistant to sampling variation.

  14. Standart Scores • Look at your worksheets and compare Tarık’s performance in Sociology and Psychology exams. What do you think? His performance is same in both exams? • In fact, it is not easy to compare two different distributions. That is like trying to compare apples and oranges. • In fact, we studied on a way that we can use to compare Tarık’s performance in both exams. Which one is that?

  15. Standart Scores • The percentile ranks are good for such comparisons, but ranks are not scores. So, we can not use them in complicated mathematical computations. • The other way is to compute standarized score. By this way, we can make both distributions same in terms of their mean and SD.

  16. Standart Scores • Computation of standart scores is easy. What we need to know is mean and the SD of the distribution. • So, a standart score is • z= X – mean / SD • Now, compute z scores for each interval. Remember, you need to use midpoints.

  17. Properties of Standart Scores • The mean of z scores is always 0 • The standart deviation of z scores is always 1 • Even though z transformation changes mean and standart deviation of the distribution, it doesn’t change its shape

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