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Measures of and Measures of Dispersion

SLWK 381 Foundations of Social Work Research II Rachel C. Casey, MSW Agenda

● Review Homework #2 ● Learn about measures of central tendency ● Practice 1 ● Break ● Learn about measures of dispersion ● Practice 2 ● Instructions for Homework #3 Measures of Central Tendency

Measures of Central Tendency = Descriptive that use a single number to summarize data level by describing what is typical in a distribution ● ● ● Levels of Chart

Level of measurement determines which measures of central tendency can be used

Nominal Ordinal Interval Ratio Mode X X X X Median X X X Mean X X Mode

● Appropriate for data at the nominal level and above ● Refers to the category with the largest frequency in the distribution ● Distributions can have more than one mode ○ Bimodal ○ Multimodal ● Does not provide information about variation in scores

How to find the mode: Identify the most frequently reported response Mode: Example

How to find the mode: Identify the most frequently reported response

TELEVISION SHOW ƒ The Bachelorette 4 Scandal 6 Mode = Scandal Brooklyn 99 4

Modern Family 3

Law & Order: SVU 3 Median

● Appropriate for data at the ordinal level and above ● Refers to the score that divides the distribution into two equal parts with half of the cases above it and half the cases below it ● Not affected by outlier scores

How to find the median: Arrange responses from lowest to highest If N is odd, the median is the number in the middle of the list If N is even, the median is the average of the two middle numbers Median: Example

How to find the median: Arrange responses from lowest to highest If N is odd, the median is the number in the middle of the list If N is even, the median is the average of the two middle numbers

Example 1: 1, 2, 3, 4, 5, 6, 7, 8, 9 Median = 5

Example 2: 10, 35, 27, 9, 18, 21 9, 10, 18, 21, 27, 35 (18+21)/2 = 19 Median = 19 Mean

● Appropriate for data at the interval/ratio level ● Refers to the arithmetic average obtained by adding up all the scores and dividing by the number of cases

“sigma” = summation “X bar” = mean

How to find the mean: X̄ = ∑X / N

“N” = number of cases Mean: Example

How to find the mean: X̄ = ∑X / N

Example 1: 1, 2, 3, 4, 5, 6, 7, 8, 9 X̄=(1+2+3+4+5+6+7+8+9)/ 9 Mean = 5 Example 2: 10, 35, 27, 9, 18, 21 X̄=(10+35+27+9+18+21)/ 6 Mean = 20 Practice 1 : Measures of Central Tendency

● Refer to the “raw data” in this Google Spreadsheet. ● For each variable in the , determine which measures of central tendency are appropriate to use and calculate them. ● Report your answers on this Google Form. Using SPSS to Calculate Measures of Central Tendency Mean: Influenced by Outliers

Outlier = a score that is very different from most others

Example: 10, 35, 27, 9, 18, 21, 578 X̄=(10+35+27+9+18+21+578)/ 7 Mean = 99.7

Sometimes, the researcher will make a judgment call to exclude outliers when calculating the mean 10, 35, 27, 9, 18, 21 X̄=(10+35+27+9+18+21)/ 6 Mean = 20 Measures of Dispersion

Measures of Central Tendency = that indicate how much variability exists in the distribution of a particular variable ● ● ● ● Standard Levels of Measurement Chart

Measures of dispersion can only be calculated for variables at the interval/ratio level

Nominal Ordinal Interval Ratio Range X X Variance X X Standard X X Deviation Range

● Appropriate for data at the interval/ratio level ● Refers to the total number of possible values between the minimum and maximum values in a distribution

How to find the range: Maximum Score - Minimum Score*

Example:

10, 35, 27, 9, 18, 21 35 - 9 = 26 Range = 26

*Rubin text identifies a slightly different formula for calculating the range, but this one is most common Range: Influenced by Outliers

● Like the mean, the range will be highly influenced by outliers

How to find the range: Maximum Score - Minimum Score

Example: Range = 569 10, 35, 27, 9, 18, 21, 578 578 - 9 = 569

10, 35, 27, 9, 18, 21 35 - 9 = 26 Range = 26 Interquartile Range

● Range based on intermediate scores, not extreme scores

○ Addresses issue of outliers ● Refers to the difference between the values of the upper quartile and lower quartile when the data is ordered and divided into four quartiles

For this class, you don’t need to know how to calculate the interquartile range. It is important to understand it as a concept. Variance

● Appropriate for data at the interval/ratio level ● Refers to the average variability in the distribution of a variable

○ In technical terms, refers to the average of the squared deviations from the mean

2 How to find the variance: Variance = ∑(X - X̄) / (N-1)

Don’t freak out! We’ll break this down! Calculating the Variance

2 How to find the variance: Variance = ∑(X - X̄) / (N-1)

2 X X̄ (X - X̄) (X - X̄) 9 20 -11 121 10 20 -10 100

18 20 -2 4 ∑(X - X̄)2 = 121+100+4+1+49+225 = 500 21 20 1 1 500/(N-1) = 500/(6-1) = 500/5 = 100 27 20 7 49 35 20 15 225 Variance = 100

● Appropriate for data at the interval/ratio level ● Refers to the average variability in the distribution of a variable measured in the same units of measurement as the original data

○ In technical terms, refers to square root of the variance

How to find the standard deviation: 2 Standard Deviation = [∑(X - X̄) / (N-1)] = Variance Calculating the Standard Deviation

2 How to find the standard deviation: [∑(X - X̄) / (N-1)]

X X̄ (X - X̄) (X - X̄)2 ∑(X - X̄)2 = 121+100+4+1+49+225 = 500 9 20 -11 121 10 20 -10 100 500/(N-1) = 500/(6-1) = 500/5 = 100

18 20 -2 4 100 = 10 21 20 1 1

27 20 7 49 Standard Deviation = 10

35 20 15 225 Why do we care about Standard Deviation?

● Standard deviation shows us how “spread out” the data is

○ Do the study participants have similar or varying experiences? ● The larger the standard deviation, the more variability in the data

Shafer, K., Fielding, B., & Wendt, D. (2017). Similarities and differences in the influence of paternal and maternal depression on adolescent well-being. Social Work Research, 41(2). Practice 2 : Measures of Dispersion

● Refer to the “raw data” in this Google Spreadsheet. ● Calculate the range, variance and standard deviation for the variables “Age” and “Weight.” ● Report your answers on this Google Form. Using SPSS to Calculate Measures of Dispersion