Variability Ibrahim Altubasi, PT, PhD The University of Jordan

Variability Look at the following two population distributions of student scores on a 5-point scale 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 1, 1, 2, 2, 3, 3, 4, 4, 5, 5 Your task: Calculate the means of the two distributions Think: Does the mean tell the difference between the two distributions?

Descriptive Statistics Descriptive Statistics: statistical procedures used to summarize, organize, and simplify data. Descriptive Statistics: statistical procedures used to summarize, organize, and simplify data. Shape of Distribution Central Tendency Variability Range Interquartile Range Variance Standard Deviation Range Interquartile Range Variance Standard Deviation Variability provides a quantitative measure of the degree to which scores in a distribution are spread-out or clustered together

Variability Variability Range Interquartile Range Variance Standard Deviation Range Interquartile Range Variance Standard Deviation The distance between the largest score and the smallest score

Variability Variability Range Interquartile Range Variance Standard Deviation Range Interquartile Range Variance Standard Deviation Interquartile range (IQR) is the distance between Q1 and Q3. Semi-Interquartile range is one half of the IQR. The 1st quartile (Q1) is the score at the 25th percentile (the score separates the lowest 25% of the distribution from the rest. The 2nd quartile (Q2) is… The 3rd quartile (Q3) is the score that divides the bottom three quarters of the distribution from the top quarter (the score at the ?? percentile)

Variability Find the range and the semi-interquartile range of the following data set: 3, 5, 3, 2, 8, 4, 6, 7, 1, 4, 3, 2

Variability Variability Range Interquartile Range Variance Standard Deviation Range Interquartile Range Variance Standard Deviation Variance and Standard deviation (SD) are the most commonly used and most important measure of variability. They use the mean as a reference and measure variability by considering the distance between each score and the mean. The SD approximates the average distance from the mean.

Variability in the Population Population Mean μ = Σ X N Deviation is the distance from the mean: Deviation Score = X- μ Population Sample Size Xμ Deviation Σ2280 μ38 Questions: Can we use the average of the deviation scores as a measure of variability? How can we represent the average distance from the mean?

Variability in the Population Sum of squared deviations, also called Sum of squares (SS): SS = Σ (X – μ) 2 = Σ(X 2 ) – (ΣX) 2 /N Xμ DeviationSquared Deviation Σ μ38 N6 Squared Deviations (X – μ) 2 SS: Σ(X – μ) 2 Question: Is SS a good measure for variability? Why or why not?

Variability in the Population Variance is the average of the squared deviations. Population variance is denoted as σ 2. σ 2 = SS/N = Σ (X – μ) 2 /N. Xμ DeviationSquared Deviation Σ μ38105 N6 Squared Deviations (X – μ) 2 SS: Σ(X – μ) 2 σ 2 = Σ (X – μ) 2 /N

Variability in the Population Standard deviation (SD) is the square root of the variance. It has the same unit as the raw score. Population SD is denoted as σ. σ = √(Σ (X – μ) 2 /N). Xμ Deviati on Squared Deviation Σ μ38105 N6 SS: Σ(X – μ) 2 σ 2 = Σ (X – μ) 2 /N

Variability in the Sample The population of adult heights forms a normal distribution. If you select a sample from this population, you are most likely to obtain individuals who are near average in height. As a result, the scores in the sample will be less variable (spread out) than the scores in the population

Variability of the Sample Standard deviation (SD) and Variance for the samples Sample Mean: = ΣX n Deviation Score = X – Sum of Squares from the sample SS = Σ (X- ) 2 = ΣX 2 – (ΣX) 2 n Sample Variance (denoted as s 2 ): s 2 = SS n-1 Sample SD (denoted as s): s= √ SS n-1

Variability The sample variance s 2 = SS / (n-1) is an unbiased estimate of the population variance. A sample statistic is unbiased if the average value of the sample statistics, obtained over many different samples, is equal to the population parameter. A sample statistic is biased if the average value of the sample statistics, obtained over many different samples, consistently underestimates or overestimates the population parameter. Sample mean ( is an unbiased estimate for population mean(μ). Sample variance (s 2 ) calculated through SS/(n-1) is an unbiased estimate for population variance (σ 2 )

Variability Adding a constant to each score will change the mean but will not change the SD or variance. Multiplying each score by a constant will cause the mean and the SD to be multiplied by the same constant, and will cause the variance to be multiplied by the square of the constant.

Variability Add or subtract a constant from each score in the data …; s 2 = 1.47 s = 1.21 s 2 = ? s = ?

Variability Multiplying or dividing each score by a constant in the data… s 2 = 2.25 s = 1.58 s 2 = ? s = ?

Descriptive Statistics Mean is the most common measure of central tendency; SD (and related variance) is the most common measure of variability. When median is used to report central tendency, interquartile range (IQR) or semi-interquartile range is commonly used to report variability. SIQR provides the best measure of variability for distributions that are very skewed or that have a few extreme scores.