Basic Statistics Measures of Variability

Measures of Variability The Range Deviation Score The Standard Deviation The Variance

STRUCTURE OF STATISTICS TABULAR Continuing with numerical approaches. DESCRIPTIVE GRAPHICAL NUMERICAL STATISTICS NUMERICAL CONFIDENCE INTERVALS INFERENTIAL TESTS OF HYPOTHESIS

STRUCTURE OF STATISTICS NUMERICAL DESCRIPTIVE MEASURES TABULAR CENTRAL TENDENCY DESCRIPTIVE GRAPHICAL NUMERICAL VARIABILITY SYMMETRY

STRUCTURE OF STATISTICS NUMERICAL DESCRIPTIVE MEASURES CENTRAL TENDENCY RANGE VARIABILITY VARIANCE NUMERICAL SYMMETRY STANDARD DEVIATION

Do you think your project will succeed ? We need the variability of IQs in the class! You are an elementary school teacher who has been assigned a class of fifth graders whose mean IQ is 115. Because children with IQ of 115 can handle more complex, abstract material, you plan many sophisticated projects for the year. Do you think your project will succeed ? General population 85% 55 70 85 100 115 130 145

Can you conclude that the pay for two occupations is equal? Is the average salary enough? We need the variability! Having graduated from college, you are considering two offers of employment. One in sales and the other in management. The pay is about same for both. After checking out the statistics for salespersons and managers at the library, you find that those who have been working for 5 years in each type of job also have similar averages. Can you conclude that the pay for two occupations is equal? Sales management Much more Much less $20,000

Single score Group of scores Central Tendency measures Mean IQ=118 IQ of 100 students

? ? ? More homogeneous Central Tendency Measures Measures of Central Tendency do not tell you the differences that exist among the scores

Same Mean---Different Variability So What? 4 How many are out here? 2 1 3 60 Central Tendency

1. The Range = The difference between the largest (Xmax) and the smallest (Xmin). 25, 21, 22, 23, 28, 26, 24, 29 21 22 23 24 25 26 28 29 Range = 29 –21 = 8 A large range means there is a lot of variability in data.

? Range = 29 –10 = 19 Drawbacks of The Range Range = 29 –26=3 10, 28, 26, 27, 29 ? Range = 29 –26=3 10 26 27 28 29 Range = 29 –10 = 19 The Range depends on only the two extreme scores

Range and Extreme Observations Because the range is determined by just two scores in the group, it ignores the spread of all scores except the largest and smallest. One aberrant score or outlier can be greatly increase the range

Before you determine the Range, all scores must be arranged in order Range and Measurement Scales Before you determine the Range, all scores must be arranged in order Country Code SES F Age 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 3-1=2 3-1=2 3-1=2 3-1=2 1=American 2=Asian 3=Mexican 1=Upper 2=Middle 3=Lower

Differences among Scores 3. The Variance Differences among Scores 78 49 49 41 88 35 66 53 44 66 49 27 83 53 95 27 Differences among Scores 81 35 42 62 72 57 41 81 49 49 63 35 41 53 ? 77 35 78 88 66

How can you determine the variability of each individual in group? Total Variability = Sum of Individual Variability How can you determine the variability of each individual in group? 72 22 22 55 70 3 12 67 The amount of Individual difference entirely depends on comparison criteria.

Can you figure out total amount of differences among scores ? Can you figure out how much each score is different from other scores ? Can you figure out total amount of differences among scores ?

You need a Common Criteria for computing Total Variability 46 48 47 Mean Score 53 ? 49 Reference score 51 50 45 52

You need a Common Criteria for computing Total Variability -3 46 -1 48 -2 47 Deviation Scores 49 +4 53 ? 49 Reference score +2 51 50 +1 45 -4 +3 52 A Deviation score tells you that a particular score deviate, or differs from the mean

DEVIATION SCORE = (Xi - Mean) A score a great distance from the mean will have large deviation score. 2 1 3 Mean A B C D E F

Sum of Deviation Scores Total amount of variability?! Sum of all distance values! mathematically No way! conceptually

The idea makes sense…but If you compute the sum of the deviation scores, the sum of the deviation scores equals zero! Sum of Deviation scores = (-4) + (-3) + (-2) + 0 + (1) + (2) + (3) + (4) = 0

The Sum of Absolute Deviation Scores The sum of absolute deviations is rarely used as a measure of variability because the process of taking absolute values does not provide meaningful information for inferential statistics.

Sum of Squares of deviation scores “SS” Conceptually And Mathematically

SS= Sum of Squares of Deviation Scores, SS Instead of working with the absolute values of deviation scores, it is preferable to (1) square each deviation score and (2) sum them to obtain a quantity know as the Sum of Squares. SS=(-4)+(-3)+(-2)+(-1)+0+(1)+(2)+(3)+(4) =16+9+4+1+0+1+2+9+16 =60 2 2 2 2 2 2 2 2 2 SS= i

So ! Group of scores “B” Group of scores “A” SS(A)=30 SS(B)=40 Can you say that the variability of the data in Group B is greater than the data in Group A?

3, 4 3, 4 =.50 What happens to SS when we look at some data? Group A Group B Mean = 3.5 Mean = 3.5 2 2 2 2 SS = (3 - 3.5) + (4 - 3.5) =.50 SS = (3 - 3.5) + (4 - 3.5) + (3 - 3.5) + (4 - 3.5) =1.00 2 2

SS tends to increase as number of data(N) increase. Mean i i=1 SS tends to increase as number of data(N) increase. SS is not appropriate for comparing variability among groups having unequal sample size. How can you overcome the limitation of SS

The resulting value will be Mean of the Deviation Scores (Mean Square) If SS is divided by N The resulting value will be Mean of the Deviation Scores (Mean Square) VARIANCE

3, 4 3, 4 Group A Group B Mean = 3.5 Mean = 3.5 2 2 2 2 V = (3 - 3.5) + (4 - 3.5) =.50/2 = .25 V = (3 - 3.5) + (4 - 3.5) + (3 - 3.5) + (4 - 3.5) = 1.00/4 = 2.5 2 2

Variance Population Variance Sample Variance

POPULATION VARIANCE Individual value Population mean Sigma Square Population size

SAMPLE VARIANCE Sample variance Sample size-1 Degree of freedom Sample Mean Individual value Sample variance Sample size-1 Degree of freedom The sample variance (S2) is used to estimate the population variance (2)

Why n-1 instead n ?

population 100 = Sampling n Sampling error sample ?<100<?

Ideally, a sample variance would be based on (x - m)2 Ideally, a sample variance would be based on (x - m)2. This is impossible since m is not known if one has only a sample of n cases. m is substituted by . The value of the squared deviations is less from X than from any other score . Hence, in a sample, the value of (X-X) n would be less than n. > n n

> Ideal sample variance One could correct for this bias by dividing by a factor somewhat less than n

7 sample n=5 We know that ? must equal 25. If we know that the mean is equal to 5, and the first 4 scores add to 18, then the last score MUST equal 7. We know that ? must equal 25. n-1 are free to change Degree of freedom

4. Standard Deviation SD Positive square root of the variance Population Sample

The Standard Deviation and the Mean with Normal Distribution

Relationship between m and s Normal Distribution m m-3s m-2s m-1s m+1s m+2s m+3s Relationship between m and s

S 68% 95% 99.9% Relationship between and Normal Distribution -3S -2S

EMPIRICAL RULE For any symmetrical, bell-shaped distribution, approximately 68% of the observations will lie within  1 standard deviation of the mean; approximately 98% within  2 standard deviations of the mean; and approximately 99.9% within  3 standard deviations of the mean.

You can approximately reproduce your data! If a set of data has a Mean=50 and SD=10, then… 68% 95% 99% 20 30 40 50 60 70 80