An Introduction to Descriptive Statistics (Will Begin Momentarily) Jim Higgins, Ed.D.

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1 An Introduction to Descriptive Statistics (Will Begin Momentarily) Jim Higgins, Ed.D. Visit BCGi Online While you are waiting for the webinar to begin, Don t forget to check out our other training opportunities through the BCGi website. Join our online learning community by signing up (it s free) and we will notify you of our upcoming free training events as well as other information of value to the HR community. 1

2 HRCI Credit BCG is an HRCI Preferred Provider CE Credits are available for attending this webinar Only those who remain with us for at least 80% of the webinar will be eligible to receive the HRCI training completion form for CE submission i About Our Sponsor: BCG Assisted hundreds of clients with cases involving Equal Employment Opportunity (EEO) / Affirmative Action (AA) (both plaintiff and defense) EEO Litigation Support / OFCCP (federal contracting) Audit Support Compensation Analyses / Test Development and Validation Published: Adverse Impact and Test Validation, 2 nd Ed., as a practical guide for HR professionals Editor & Publisher: EEO Insight an industry e-journal Creator and publisher of a variety of productivity Software/Web Tools: OPAC (Administrative Skills Testing) CritiCall (9-1-1 Dispatcher Testing) AutoAAP (Affirmative Action Software and Services) C 4 (Contact Center Employee Testing) Encounter (Video Situational Judgment Test) Adverse Impact Toolkit (free online at AutoGOJA (Automated Guidelines Oriented Job Analysis ) Industry Leader 4 2

3 An Introduction to Descriptive Statistics Jim Higgins, Ed.D. Jim Higgins, Ed.D. Biddle Consulting Group, Inc. 193 Blue Ravine, Ste. 270 Folsom, CA

4 Overview of Presentation All Numbers Are NOT Created Equal Gender Ethnicity IQ Test Score Average=0.67 Average=2.33 Average=92.89 Average=

5 Some Numbers Convey More Information Than Others Can t do Complex Math Can do Complex Math Levels of Measurement Interval and Ratio Scales Nominal and Ordinal Scales 5

6 Measurement Scales Counts (Frequencies) Percentages NO complex math Nominal Ordinal Counts (Frequencies) Percentages Place into a logical or relative eatve ode order Counts (Frequencies) Percentages Order Complex Math Scale Special Considerations Binary or Dichotomous Items Ordinal Variables Under certain circumstances you CAN use these as special cases of Scale Variables 6

7 What Are Descriptive Statistics Organize Summarize Synthesize Which Is Easier To Understand? Gender Age Test Score Male Male Female Male Female Female Female Male Male Male Male Female Male Male Sample Size = 14 Gender Male = 9 (64.3%) Female = 5 (35.7%) Average Age = 33 Males = 31.4 Females = 34.6 Average Test Score = 76.5 Males = 74.3 Females =

8 So Easy to use Most appropriate for use to communicate with nonstatisticians and for management Can be hard to use well Types of Descriptive Statistics Tabular Frequency Tables Graphical Pie Charts Bar Charts Histograms Scatter Plots Statistical Measures of Central Tendency Measures of Variability 8

9 Frequency Tables Counts/Frequency of each score/response Percent Valid Percent Cumulative Percent Graphs Can Be Important in Helping People To Understand Time to Re offense by County Program County A County B County C Test Scores by Administration 0 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr African American White Asian/Pacific Islander 0 Admin 1 Admin 2 Admin 3 Admin 4 9

10 Important! Bar Charts and Pie Charts work best with: Variables measured on nominal OR ordinal scales When your variable has fewer than 10 or so categories When Good Bar Charts Go Bad Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. East West North South Mid-West 10

11 Pie Chart s Gone Wrong Age at Time of First Contact With Law Enforcement Measures of Central Tendency 11

12 What is a Measure of Central Tendency? A single summary number that tells you about what is most common. Gives you an idea about what is the most common number in a distribution of scores. Examples of Central Tendency Most scores fall in this central area Most scores fall in this central area Skew! 12

13 Three Basic Measures of Central Tendency Mode Mdi Median Mean The Mode Suppose you asked 10 people their ages and you got: 19, 19, 20, 21, 21, 21, 21, 22, 23, 24, Notice that there are more 21s than any other single number? In this case, 21 is the most frequently occurring score. Mode = 21 13

14 More Mode Examples 7, 10, 11, 11, 12, 17 Mode = , 111, 156, 177, 180 Mode = , 6.23, , 7.00 Mode = ,.172,.299, 3.00, 3.00 Mode =? There are two modes!.172 and This is called bi-modal. The Median 50% 50% The Median 14

15 The Median: An Example Suppose you had the following 10 scores: Scores are Higher The Median = 9 5 Scores are Lower More Median Examples 7, 8, 9, 10,11 Median = 9 5, 10, 15, 20 Median = 12.5 (15 10)/2 100, 135, 135, 135, 135 Median =

16 The Mean The most important measure of central tendency. Also known as The Average The Mean The formula! n 16

17 The Mean Break n it down This This n n is the symbol for the mean. symbol means Sum of and simply tells you to add things up refers to a variable. (A lower case N ) refers to the number of s that you have. Reading the formula n The mean equals The sum of the x scores Divided by the number of x scores 17

18 The Mean: Working an example: Suppose you asked 5 people their ages and got the following numbers: 19, 24, 23, 21, 20 What is the mean? n = 107 n = 5 Calculating the Mean n

19 A few Mean Examples 21, 35, 36, 27, / 5 = , 5.1, 8.26, / 4 = , 133, 174, 211, / 5 = Summary: Measures of Central Tendency There are 3 measures of central tendency: Mode: Most Frequently occurring number Median: Number where 50% are higher and 50% are lower Mean: The arithmetic Average 19

20 Important! Each measure of central tendency is looking at what is most common about a distribution but Each takes a different perspective! Check This Out! If we have five people aged 12, 12, 15, 17, 45 What is the Mode? What is the Median? What is the Mean? The Mode is 12 The Median is 15 The Mean is

21 Skewness Normal Negative Skew Positive Skew A Picture is Worth In a normal distribution: Mode = Median = Mean Mode In a Skewed distribution, all bets are off! Median Mean 21

22 So What? The Mode ALWAYS means the same thing the the most common score. The Median ALWAYS means the same thing 50% higher and 50% lower. The Mean ONLY MAKES SENSE WHEN YOU HAVE A NORMAL DISTRIBUTION. This has significant implications to interpreting test and other statistics! What is Variation? Refers to how spread out scores are around the mean Tells you whether most scores are close to the mean or if they are really a lot different Small variation means most scores are close to the mean Large variation means many of the scores are much higher or much lower than the mean 22

23 Variation: Examples Little variation: Most scores are close to the mean. Moderate variation: While most scores tend to be close to the mean, some are farther away. Large variation: A lot of the scores are far away from the mean. Importance of Measures of Variation Along with the mean, it helps you get a fuller picture of what is happening with the data. Helps you put the mean in context. When you make predictions about scores, you are able to tell how accurate your predictions are likely to be. 23

24 Four Important Measures of Variation Least Important Most Important Range Sum of Squares Variance Standard Deviation The Range Highest Score Lowest Score Range Suppose you asked 5 people their hourly salary and you collected the following data: Respondent Salary $10 $12 $13 $24 $25 $25 (The highest score) $10 (The lowest score) $15 (The range) 24

25 Shortcomings of the Range While it is easy to compute, it provides limited information. It is difficult to compare the amount of variability in different distributions because the range is so dependent on the scale you are measuring things on. Sum Of Squares 2 In English: The sum of the squared deviation scores I Still Don t Get It: The sum of the squared differences of each score from the mean O.K., I may be dense but I still don t get it! Subtract the mean from each raw score and square the result. Once you have done it for each score, add them all up. 25

26 Breakin Down The Sum Of Squares 2 Subtracting the mean from a raw score results in a deviation score. This tells you how much that score deviates from the mean. When you square a deviation score, you get a squared deviation score! 2 When you add up the squared deviation scores, you get The sum of the squared deviation scores. The Sum of Squares! Calculating the Sum of Squares Salary Deviation Scores Squared Deviation Scores $10 $12 $13 $24 $ = Σ = 0 Σ = This is the Sum of Squares or, the sum of the squared deviations of each score from the mean. 26

27 Problems With The Sum of Squares The sum of squares suffers from the same problems as the range. Hard to interpret in a meaningful way Difficult to make comparisons between the variability of different distributions based on the sum of squares. The Variance sd 2 ( ) n

28 The Variance (Continued) Sum of Squares sd 2 ( n 1 ) 2 The number of people in your set of data minus 1 Calculating the Variance Salary $10 $12 $13 $24 $25 Deviation Scores Squared Deviation Scores = 16.8 Σ = 0 Σ = Variance = / / 4 = 50.7 Sum of Squares 28

29 What is the variance? Point 1 The sum of squares is the sum of the squared deviations of each score from the mean. Point 2 The variance is the sum of squares divided by n Therefore, the variance is the average of the squared deviations of each score from the mean. Problems with the Variance Generally, the same problems as the Range and the Sum of Squares Hard to interpret Hard to make comparisons Another important problems is that it is reported in squared units (because you had to square deviation scores to get it) 29

30 The Standard Deviation The square root of the variance sd ( ) n 1 2 When you take the square root of the variance, you change the variance from being squared deviation scores back into raw scores. The standard deviation tells you, on average, how much scores deviate from the mean. Example 16.8 Variation Range = 15 Sum of Squares = Variance = 50.7 Standard Deviation =

31 Example 34% 34% 2% 14% 14% 2% The Z-Distribution Example 2% 14% 34% 34% What percent of people have scores that are lower than 1 standard deviation above the mean? 31

32 Example 14% 2% What percent of people have scores that are higher than 1 standard deviation above the mean? Example 2% 14% 34% What percent of people have scores that are lower than the mean? 32

33 Example 34% 34% 14% What percent of people have scores fall between -1 and +2 standard deviations around the mean? Example 14% 34% What percent of people have scores fall between -2 standard deviations below the mean and the mean? 33

34 Summary We talked about 4 measures of variation Range Sum of Squares Variance Standard Deviation The most important and useful of these is the Standard Deviation Consider the Implications! The mean is sensitive to skewness. It is pulled in the direction of the skew. In a skewed distribution, the mean may not even effectively measure the distribution s central tendency (i.e., tell you what is most common). Since the mean is used to calculate most other important statistics, if the mean is not accurate, then all statistical results may be open to question. The importance of understanding the mean and standard deviation cannot be overstated! 34