Chapter 4 of Pagano’s Understanding Statistics in the Behavioral Sciences Measures of Central Tendency and Variability Presented by David R. Dunaetz Azusa Pacific University

Quiz #2 Is this a frequency distribution table? Is this a bar chart? Is this a histogram?

The Mean (Average) Add up all the raw scores and divide by the number of raw scores.

Calculating the Mean Application: How often do students who are named after places in France (who sit around discussing food) have frozen burritos for lunch on the average?

Predicting the mean of a population. = mean of a sample. Our best estimate of μ is

Calculating the mean in an Excel table Method 1: AutoSum, Average AutoSum may be found under Home or Formulas Place the curser at the bottom of a column of data in a table. Click AutoSum Choose Average

Other ways of calculating the mean In any cell, put a formula =average(b2..b26) =average(b2:b26) =average(Table1[Bieber])

The Median The middle score (if you arrange the scores in order). If there’s an even number of scores, there will be no middle score. So take the average of the middle 2 scores. Pretty similar to P50

The Mode Mode = style in French The most common score. Distributions can be bimodal (2 modes). “Camel” distributions.

Mean vs. Median vs. Mode The (Arithmetic) Mean The Median The Mode The only one that can be calculated algebraically. The only one where every score counts. The only one sensitive to extreme scores The Median Not sensitive to extreme scores. The Mode Easiest to see in frequency curves. Not always meaningful. Example: The Weight of Bears Beep-Beep: 400g Oliver: 300g Ugly: 50g

Using Data Analysis to calculate measures of Central Tendency Make sure you have installed the Data Analysis ToolPak (Download Chapter 4 Justin Bieber example.xslx) Go to Data, Data Analysis Choose Descriptive Statistics, OK Choose input range, grouped by columns, with Label in first row. Choose a cell for the output range. Choose summary statistics. Fix up presentation

Overall Mean Example One night, the three bears sneak into a bakery. . . Ugly: 5 cookies , average weight 10g. Oliver: 2 cookies, average weight 15g. Beep-Beep: 12 cookies, average weight 100g What was the average weight of the cookies eaten by the 3 bears?

The Overall (Weighted) Score/Mean When we have the averages or scores ( ) for several (k) groups, we can find the overall average if we know how many people or items are in each group (ni).

Weighted Scores on Excel Suppose we’re evaluating a candidate based on several criteria, for each of which we have a score on a 0-100 scale. Work Experience: 72 Education: 92 Interview 1: 91 Interview 2: 61 Cognitive Ability Test 98 Integrity Test 80 The first four are worth 20% each, and the two tests are worth 10% each.

Weighted Scores: SUMPRODUCT function. Weighted average = SUMPRODUCT (Weight, Scores)/Total Weight = SUMPRODCIT (B2..B7, C2..C7)/B9 = SUMPRODUCT(Table1[Weight],Table1[Score])/B9

Weighted Scores: More than One Candidate If we had more than one candidate, we could calculate a weighted score for each one: Candidate 1: 81.0 Candidate 2: 81.5

Measures of Central Tendency and Symmetry

Two Types of Measures of Any Distribution Measures of Central Tendency Mean (average) Median (middle) Mode (most popular) Measures of Variability: How spread out the data is (the opposite of consistency) Range (high – low) Standard Deviation (expected variation) Variance (square of standard deviation)

Range Range = Lowest Score through Highest Score Highest Score – Lowest Score Weight of bears: 50-400 grams or . . . 400 – 50 grams = 350 grams Only based on extreme scores. Is there some measure of variability that would take into account each score (for example, the number of frozen burritos eaten last week by each girl)? Example: The Weight of Bears Beep-Beep: 400g Oliver: 300g Ugly: 50g

How to Measure Variation? What would measure variation?

Standard Deviation A deviation score tells how far a raw score is from the mean. X - X = deviation from the mean Now Σ (X-X) = 0, so averaging the deviation scores will always give a big, fat 0. Solution: Square the deviations before averaging them! (Squared numbers are always positive.) Σ (X-X)2 is called the sum of the squared deviations, or the Sum of the Squares or SS

Standard Deviation (cont.) (Σ (X-X)2)/N gives the average of squared deviations. (Σ (X-X)2)/N gives strange units, like burritos squared. So take the square root. We call this σ (standard deviation of a population) if we have data for the whole population. To calculate the standard deviation from a sample (s) in order to estimate σ, we use a slightly different equation. . .

Standard Deviation (for a sample to estimate the standard deviation of the population) Another way of writing this formula:

Excel Formulas for the SD In any cell, put a formula =STDEV(b2:b26) =STDEV.S(b2:b26) =STDEV(Table1[Bieber]) Can also be calculated by AutoSum (choose StdDev or STDEV). Also included in Data Analysis (Descriptive Statistics) Note: You don’t ever need to use STDEVP or STDEV.P which calculates the SD with N (vs. N-1) in the denominator. Only for entire populations

Everyday Meaning of the Standard Deviation. One standard deviation around the mean: most scores. 68% Two standard deviations around the mean: almost all scores. 95% Example: Average test score is 85 with a standard deviation of 10. Most scores are between 75 and 95 (85 +/- 10) Almost all scores are between 65 and 105 (85 +/- 20)

Variance Variance = s2 = estimated σ2 (population variance) Main advantages over the standard deviation: No square root needed Can be added together or subtracted (useful for ANOVA, analysis of the variance)

Data Analysis Toolpak: Descriptive Statistics Go to Data, choose Data Analysis Choose Descriptive Statistics, OK Choose Input Range with Label in first row (Be careful) Choose output range Put on same worksheet. Be careful Choose Summary Statistics Format output so it looks nice