1 Univariate Descriptive Statistics Heibatollah Baghi, and Mastee Badii George Mason University

2 Objectives Define measures of central tendency and dispersion. Select the appropriate measures to use for a particular dataset.

3 How to Summarize Data? Graphs may be useful, but the information they offer is often inexact. A frequency distribution provides many details, but often we want to condense a distribution further.

4 1.Measures of Central Tendency. 2.Measures of Variability or Scatter. Two Characteristics of Distributions

5 Measures of Central Tendency: Mean The mean describes the center or the balance point of a frequency distribution. The sample mean: Calculate the mean value for the following data: 23, 23, 24, 25, 25,25, 26, 26, 27,

6 Measures of Central Tendency: Mode The most frequent value or category in a distribution. Calculate the mode for the following set of values: 20, 21, 21, 22, 22, 22, 22, 23, 23,

7 Measures of Central Tendency: Median The middle value of a set of ordered numbers. Calculate for an even number of cases. 21, 22, 22, 23, 24, 26, 26, 27, 28, Calculate for odd number of data with no duplicates: 22, 23, 23, 24, 25, 26, 27, 27, Median changes when data at center repeats.

8 Comparison of Measures of Central Tendency Mode Most frequently occurring value Nominal, Ordinal, and (sometimes) Interval/Ratio-Level Data Median Exact center (when odd N) of rank-ordered data or average of two middle values (when even N) Ordinal-Level Data and Interval/Ratio-Level data (particularly when skewed) Mean Arithmetic average (Sum of Xs/N) Interval/Ratio-Level Data

9 Comparison of Measures of Central Tendency in Normal Distribution Mean, median and mode are the same Shape is symmetric

10 Comparison of Measures of Central Tendency in Bimodal Distribution Mean & median are the same Two modes different from mean and median

11 Comparison of Measures of Central Tendency in Negatively Skewed Distributions Mean, median & mode are different Mode > Median > Mean Outliers pull the mean away From the median

12 Comparison of Measures of Central Tendency in Positively Skewed Distributions Mean, median & mode are different Mean > Median > Mode Outliers pull the mean away From the median

13 Comparison of Measures of Central Tendency in Uniform Distribution Mean, median & mode are the same point

14 Comparison of Measures of Central Tendency in J-shape Distribution Mode to extreme right Mean to the right of median

15 Measures of Variability or Scatter Reporting only an average without an accompanying measure of variability may misrepresent a set of data. Two datasets can have the same average but very different variability.

16 Measures of Variability or Scatter: Range The difference between the highest and lowest score Easy to calculate Highly unstable Calculate range for the data: 110, 120, 130, 140, 150, 160, 170, 180, – 110 = 80

17 Measures of Variability or Scatter: Semi Inter-quartile Range Half of the difference between the 25% quartile and 75% quartile SQR = (Q3-Q1)/2 More stable than range

18 Measures of Variability: Sample Variance The sum of squared differences between observations and their mean [ss = Σ (X - M) 2 ] divided by n -1. Sample variance : Standard deviation squared Formula for sample variance

19 Measures of Variability or Scatter: Standard Deviation The squared root of the variance. Calculate standard deviation for the data: 110, 120, 130, 140, 150, 160, 170, 180, 190.

20 Calculating Standard Deviation Sample Sum of Squares: Sample Variance Sample Standard Deviation SS is the key to many statistics

21 Calculating Standard Deviation DataX-M(X - M) Total06000 (SS) N-19 Sample Variance667 Standard Deviation25.8 SS is the key to many statistics

22 Formula Variations Calculating formula Defining formula Sum of squares Variance Standard deviation

23 Comparison of Measures of Variability and Scatter In Normal Distribution Range ~ 6 standard deviation Standard Deviation partitions data in Normal Distribution

24 Standardized Scores: Z Scores Mean & standard deviations are used to compute standard scores Z = (x-m) / s Calculate standard deviation for blood pressure of 140 if the sample mean is 110 and the standard deviation is 10 Z = 140 – 110 / 10 = 3

25 Value of Z Scores Allows comparison of observed distribution to expected distribution Observed Expected

26 Take Home Lesson Measures of Central Tendency & Variability Can Describe the Distribution of Data