Comparing the Mode, Median, and Mean Three factors in choosing a measure of central tendency 1.Level of measurement –Nominal –Ordinal –Interval/Ratio 2.Shape or form of the distribution of data –Kurtosis –Skewness –Normality 3.Research Objective

Chapter 4 Measures of Variability

Chapter 4 - Introduction Measures of central tendency alone portray an incomplete picture Measures of variability allows us to understand how scores are distributed Three types – Range – Variance – Standard Deviation

Measures of Variability and Dispersion For example, two tests were given with the following results: – Test 1: – Test 2:

The Range Simplest and quickest measure of distribution dispersion Range = Difference between highest and lowest scores in a distribution In equation form: Provides a crude measure of variation Outliers severely affect the range R = range H = highest score in a distribution L = lowest score in a distribution 1, 2, 2, 4, 5, 5, 8, 9, 9, 10, 10, 10

The Inter-Quartile Range Inter-quartile range manages effects of extreme outliers In equation form: The larger the size of IQR, the greater the variability IQR = inter-quartile range Q 1 = score at the 1 st quartile, 25% below, 75% above Q 3 = score at the 3 rd quartile, 75% below, 25% above

Illustration: Range and IQR Probation Officer A – 18, 18, 19, 19, 20, 20, 22, 23 Probation Officer B – 18, 18, 19, 19, 20, 22, 43

The Variance and the Standard Deviation Deviation: distance of any given raw score from its mean Deviation Most appropriate for interval/ratio data Used to divide and discuss a normal curve Need a measure of variability that takes into account every score

The Raw-Score Formula for Variance and Standard Deviation In equation form: – Variance – Standard deviation = sum of the squared raw scores = mean squared = total number of scores

Illustration: Using Raw Scores X

X Step 1: Square each raw score and sum both columns Step 2: Obtain the mean and square it XX2X ΣX = 30ΣX 2 = / 6 = = 25 Step 3: Insert results from Step 1 and 2 into the formulas N = 6 ΣX 2 = 202 Xbar = 25

Illustration: Variance and Standard Deviation On a 20 item measure of self-esteem (higher scores reflect greater self-esteem), five teenagers scored as follows: 16, 5, 18, 9, Calculate the range 2.Calculate the variance 3.Calculate the standard deviation

The Meaning of the Standard Deviation Standard deviation converts the variance to units we can understand. But, how do we interpret? – Standard deviation represents the average variability in a distribution. – It is the average of deviations from the mean. – The greater the variability – the larger the standard deviation – Allows for comparison between a given raw score in a set against a standardized measure

The Normal Curve

Measuring the Base Line in Units of Standard Deviation when the standard deviation is 5 and the mean is 80

Variance and Standard Deviation of a Frequency Distribution # of Classf N = 20 The table on the right is a simple frequency distribution of the number of courses taken by each full time student in a particular class.

Variance and Standard Deviation of a Grouped Distribution Class Intervalf N = 25 The table on the right is a grouped frequency distribution of 25 individuals and their ages when first married.

Step 1: Find each midpoint and multiply it (m) by the frequency (f) in the class interval to obtain the fm products, and then sum the fm column. Class IntervalMidpoint (m) ffmfm (31*2) = (28* 3) = Σfm = 574

Step 2: Square each midpoint and multiply the frequency of the class interval to obtain the f(m 2 ) products, and then sum the f(m 2 ) columns. Class IntervalMidpoint (m) ffmfmf(m 2 ) , , , , , Σfm = 574Σf(m 2 ) = 13,552

Step 3: Obtain the mean and square it X m = Σ fm N X m = 574 / 25 = X m 2 = X m 2 = s 2 = (13,552/25) – s 2 = s 2 = √14.92 s = 3.86 Step 4: Calculate the variance using the results from the previous steps s = Σfm 2 N - X m 2 From the table: N = 25 Σfm = 574 Σf(m 2 ) = 13,552

Illustration: Grouped Frequency Distribution Twenty-five judges from superior courts, drug courts, and traffic courts were monitored to determine the number of decisions handed down during a particular week. For the following grouped frequency distribution, find the variance and standard deviation. Class Interval F

Selecting the Most Appropriate Measure of Dispersion It is harder to determine the most appropriate measure of dispersion than it is to determine the most appropriate measure of central tendency Not as “tied” to level of measurement Range can always be used – Regardless of data level or distribution form – Limited in information Variance and standard deviation are good for interval and some ordinal data

Comparing Measures of Variability Range is simple to calculate but not reliable The standard deviation can never be greater than the range. Standard deviation: – reflects the effect of all scores – requires data at the interval level

Summary Measures of variability allow distributions of data to be described more completely No widely accepted measures of variability for categorical data Variance and standard deviation can be used to measure deviation or dispersion within a variable