Presentation is loading. Please wait.

Presentation is loading. Please wait.

Variability Quantitative Methods in HPELS 440:210.

Similar presentations


Presentation on theme: "Variability Quantitative Methods in HPELS 440:210."— Presentation transcript:

1 Variability Quantitative Methods in HPELS 440:210

2 Agenda Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

3 Introduction Statistics of variability:  Describe how values are spread out  Describe how values cluster around the middle Several statistics  Appropriate measurement depends on:  Scale of measurement  Distribution

4 Basic Concepts Measures of variability:  Frequency  Range  Interquartile range  Variance and standard deviation Each statistic has its advantages and disadvantages

5 Agenda Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

6 Frequency Definition: The number/count of any variable Scale of measurement:  Appropriate for all scales  Only statistic appropriate for nominal data Statistical notation: f

7 Frequency Advantages:  Ease of determination  Only statistic appropriate for nominal data Disadvantages:  Terminal statistic

8 Calculation of the Frequency  Instat Statistics tab Summary tab Group tab  Select group  Select column(s) of interest  OK

9 Agenda Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

10 Range Definition: The difference between the highest and lowest values in a distribution Scale of measurement:  Ordinal, interval or ratio

11 Range Advantages:  Ease of determination Disadvantages:  Terminal statistic  Disregards all data except extreme scores

12 Calculation of the Range  Instat Statistics tab Summary tab Describe tab  Calculates range automatically  OK

13 Agenda Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

14 Interquartile Range Definition: The difference between the 1 st quartile and the 3 rd quartile Scale of measurement:  Ordinal, interval or ratio  Example: Figure 4.3, p 107

15

16 Interquartile Range Advantages:  Ease of determination  More stable than range Disadvantages:  Disregards all values except 1 st and 3 rd quartiles

17 Calculation of the Interquartile Range  Instat Statistics tab Summary tab Describe tab  Choose additional statistics  Choose interquartile range  OK

18 Agenda Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

19 Variance/SD  Population Variance:  The average squared distance/deviation of all raw scores from the mean  The standard deviation squared  Statistical notation: σ 2 Scale of measurement:  Interval or ratio Advantages:  Considers all data  Not a terminal statistic Disadvantages:  Not appropriate for nominal or ordinal data  Sensitive to extreme outliers

20 Variance/SD  Population Standard deviation:  The average distance/deviation of all raw scores from the mean The square root of the variance Statistical notation: σ Scale of measurement:  Interval or ratio Advantages and disadvantages:  Similar to variance

21 Calculation of the Variance  Population Why square all values? If all deviations from the mean are summed, the answer always = 0

22 Calculation of the Variance  Population Example: 1, 2, 3, 4, 5 Mean = 3 Variations:  1 – 3 = -2  2 – 3 = -1  3 – 3 = 0  4 – 3 = 1  5 – 3 = 2 Sum of all deviations = 0 Sum of all squared deviations Variations:  1 – 3 = (-2) 2 = 4  2 – 3 = (-1) 2 = 1  3 – 3 = (0) 2 = 0  4 – 3 = (1) 2 = 1  5 – 3 = (2) 2 = 4 Sum of all squared deviations = 10 Variance = Average squared deviation of all points  10/5 = 2

23 Calculation of the Variance  Population Step 1: Calculate deviation of each point from mean Step 2: Square each deviation Step 3: Sum all squared deviations Step 4: Divide sum of squared deviations by N

24 Calculation of the Variance  Population σ 2 = SS/number of scores, where SS =  Σ(X -  ) 2 Definitional formula (Example 4.3, p 112) or  ΣX 2 – [(ΣX) 2 ] Computational formula (Example 4.4, p 112)

25 Computational formula Step 4: Divide by N

26 Computation of the Standard Deviation  Population Take the square root of the variance

27 Agenda Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

28 Variance/SD  Sample Process is similar with two distinctions:  Statistical notation  Formula

29 Statistical Notation Distinctions Population vs. Sample σ 2 = s 2 σ = s  = M N = n

30 Formula Distinctions Population vs. Sample s 2 = SS / n – 1, where SS =  Σ(X - M) 2 Definitional formula  ΣX 2 - [(ΣX) 2 ] Computational formula Why n - 1?

31 N vs. (n – 1)  First Reason General underestimation of population variance  Sample variance (s 2 ) tend to underestimate a population variance (σ 2 )  (n – 1) will inflate s 2  Example 4.8, p 121

32 Actual population σ 2 = 14 Average biased s 2 = 63/9 = 7Average unbiased s 2 = 126/9 = 14

33 N vs. (n – 1)  Second Reason Degrees of freedom (df)  df = number of scores “free” to vary  Example: Assume n = 3, with M = 5 The sum of values = 15 (n*M) Assume two of the values = 8, 3 The third value has to be 4 Two values are “free” to vary df = (n – 1) = (3 – 1) = 2

34 Computation of the Standard Deviation of Sample  Instat Statistics tab Summary tab Describe tab  Calculates standard deviation automatically  OK

35 Agenda Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

36 When to use the frequency  Nominal data  With the mode When to use the range or interquartile range  Ordinal data  With the median When to sue the variance/SD  Interval or ratio data  With the mean

37 Textbook Problem Assignment Problems: 4, 6, 8, 14.


Download ppt "Variability Quantitative Methods in HPELS 440:210."

Similar presentations


Ads by Google