VARIABILITY

PREVIEW PREVIEW Figure 4.1 the statistical mode for defining abnormal behavior. The distribution of behavior scores for the entire population is divided into three sections. Extreme abnormal Average Normal behavior Extreme abnormal

Overview Definition Definition Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.

Overview In general a good measure of variability will serve two purposes. Variability describes the distribution. Specifically, it tells whether the scores are clustered close together or are spread out over a large distance. Variability measure how well an individual score ( or group of scores ) represents the entire distribution

The Range Range = URL X max – LRL X min. The range is the difference between the upper real limit of the largest ( maximum ) X value and the lower real limit of the smallest ( minimum ) x value.

The Interquartile Range And Semi- Interquartile Range The interquartile range is the distance between the first quartile and the third quartile : The interquartile range Q3 – Q 1

The Semi- Interquartile Range The semi- interquartile range is one-half of the interquartile range: Q3 –Q12 semi- interquartile range = Q3 – Q1 2

Standard Deviation And variance A Population Deviation is distance from the mean : Deviation score = X - μ Population variance = mean squared deviation. Variance is the mean of the squared deviation scores. Standard deviation = variance

Sum Of Squared Deviations ( ss ) Variance = mean squared deviation = SS, or sum of squares, is the sum of squared deviation scores. Definitional formula : SS = Σ ( x – μ ) Sum of squared deviations Member of scores

Formulas For Population Standard Deviation And Variance Variance = Standard deviation is the square root of variance, so the equation for standard deviation is ss N SS N

SUMMARY OF COMPUTATION FOR VARIANCE AND STANDARD DEVIATION 1. Find the distance From the mean For each individual 2. Square each distance Find the sum of the squared distance. this value is called ss or sum of squares. ( note : ss can also be obtained using the computational formula instead of steps 1 – 3 ) 4. Find the mean of the squared distance. This value called variance and measure the average squared distance from the mean. 5. Take the square root of the variance. This value is called standard deviation and provides a measure of the standard distance from the mean.

FIGURE 4.4

Graphic representation Of The Mean And Standard Deviation Figure 4.5 The graphic representation of a population with a mean of μ = 40 and standard deviation of σ = 4.

Standard Deviation And Variance For Samples Figure 4.6 the population of adult heights form a normal distribution.

Standard Deviation And Variance For Samples 1. Find the deviation for each score: deviation = x – x 2. Square of each deviation: squared deviation = ( x – x) 2 3. Sum of the squared deviation: SS =Σ ( x – x ) 2 These three steps can be summarized in a definitional formula SS : Definitional formula: SS = Σ ( x – x ) 2 Definitional formula: SS = Σ ( x – x ) 2 The value of ss can also be obtained using the computational formula using sample notation, this formula is: Computational formula : ss = ΣX 2 - ( Σ X ) 2 n

Standard Deviation And Variance For Samples Sample variance = S 2 = Sample standard deviation ( identified by the symbols ) is simply the square root of the variance. sample standard deviation = s = SS n- 1 SS n- 1

Biased And Unbiased Statistics Definitions a sample statistic is unbiased if the sample statistic. Obtained over many different samples is equal to the population parameter. On the other hand if the average value for a sample statistic consistently underestimates or consistently overestimates the corresponding population parameter, then the statistic is biased.

Biased And Unbiased Statistics Degrees of freedom or df for a sample are defined as df = n – 1 Where n is the number of scores in the sample.

Transformations Of Scale 1. Adding a constant to each score will not change the standard deviation. 2. Multiplying each score by a constant causes the standard deviation to be multiplied by the same constant.

Factors That Affect Variability 1. Extreme scores 2. Sample size 3. Stability under sampling 4. Open – ended distributions.