Measures of Central Tendency; Dispersion Seminar 3 30 slides – 45 mins; pay attention to what the mathematical symbols represent (e.g., N, M, s, theta) In tutorials, students were lost at the SD and z scores. It might be worthwhile, since there is extra time, to demonstrate how they can use Excel to calculate SD and z scores.

Measures of central tendency Part 1 Measures of central tendency

Recap: Last week Stress rating Frequency 10 14 9 15 8 26 7 31 6 13 5 4 7 7 7 8 8 7 8 9 4 7 3 6 9 10 5 7 10 6 8 7 8 7 8 7 4 5 10 10 0 9 8 3 7 9 7 9 5 8 5 0 4 6 6 7 5 3 2 8 5 10 9 10 6 4 8 8 8 4 8 7 3 7 8 8 8 7 9 7 5 6 3 4 8 7 5 7 3 3 6 5 7 5 7 8 8 7 10 5 4 3 7 6 3 9 7 8 5 7 9 9 3 1 8 6 6 4 8 5 10 4 8 10 5 5 4 9 4 7 7 7 6 6 4 4 4 9 7 10 4 7 5 10 7 9 2 7 5 9 10 7 2 5 9 8 10 10 6 8 3 Stress rating Frequency 10 14 9 15 8 26 7 31 6 13 5 18 4 16 3 12 2 1

Today’s Question How can we summarize a distribution of scores efficiently using quantitative (as opposed to graphical) methods?

Measures of Central Tendency Central tendency: most “typical” score which “best” represents the distribution Mode (b) Median (c) Mean

Measures of Central Tendency Mode: most frequently occurring score 10, 20, 30, 40, 40, 50, 60 Mode = 40

Measures of Central Tendency Median: the value at which 1/2 of the ordered scores fall above and 1/2 of the scores fall below; 50th percentile 1 2 3 4 5 1 2 3 4 Median = 3 Median = 2.5

Measures of Central Tendency Mean: The “balancing point” of a set of scores; the average Three types: Arithmetic mean (in this course, “mean” = “arithmetic mean” Geometric mean Harmonic mean Don’t need to know

Mathematical procedures Summation 1 𝑁 𝑋 𝑛 = 𝑋 1 + 𝑋 2 + 𝑋 3 Average 1 𝑁 1 𝑁 𝑋 𝑛 = 𝑋 1 + 𝑋 2 + 𝑋 3 3 n (subject ID) X (e.g., exam score) 1 19 2 27 3 23 Note: Some students need more time to understand algebraic formulas.

Mathematical procedures Sum of the squared values vs. Square of the summed values Note 1 3 𝑋 𝑛 2 ≠ ( 1 3 𝑋 𝑛 )2 1 3 𝑋 𝑛 2 =192+272+232 ( 1 3 𝑋 𝑛 )2=(19+27+23)2 n X 1 19 2 27 3 23 This is important when we want to calculate “sums of squares”

The balancing point B A C D E 3 4 5 6 7 8 9  (+1)   (-1)  (-2) (+4) (-2) (– 1) + (– 2) + (– 2) + 1 + 4 = 0

An unbalanced pivot B A C D E 3 4 5 6 7 8 9 (-1) (-3) (+2) (- 4) (- 4) (– 1) + (– 3) + (– 4) + (– 4) + 2 = –10

Another way to conceptualize “balance” At the balancing point, the sum of all the mean deviations is 0. 𝑥 −𝑀 =0 What we want to do next is solve this equation for M. We first distribute the summation operation and move one term to the right-hand side.

Next, we note that the sum of a bunch of M’s is simply the number of M’s (N) times M. If we divide both sides by N, we find that the balancing point is equal to the sum of all the scores, divided by the total number of scores.

Measures of Central Tendency Mean = 30/10 = 3

Measures of Central Tendency 10 observations Theoretically When the distribution of scores is normal, the mode = median = mean ∞ observations Mean Median Mode

Measures of Central Tendency Mode = 2 Median = 2.5 Mean = 2.7 When scores are positively skewed, mean is dragged in direction of skew and mode < median < mean When scores are negatively skewed, mean is dragged in direction of skew and mode > median > mean

Kurtosis Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. What does it mean when there is extreme… Positive kurtosis? Negative kurtosis?

Measures of Central Tendency The most commonly used measure of central tendency is the mean Why? It uses all the information in the scores Can be algebraically manipulated with ease

Dispersion (or spread) Part 2 Dispersion (or spread)

Today’s Questions In some datasets the scores are tightly clustered around the mean. In other data sets, the scores are spread out. How can we quantify this property of distributions?

Scenario You go to a fancy restaurant with a group of friends. Each person orders his/her own food. You ordered the cheapest food on the menu. There is peer pressure to split the bill evenly. Would you object to it?

Measures of Spread What is spread or dispersion? The degree to which scores are clumped around the mean. Note: If you have two datasets with identical means, you may not necessarily have the same spread of data!

How can we quantify spread? Spread: how far, on average, an individual score is from the mean. Recall: “how far an individual score is from the mean” (often called “deviation scores”) “on average” Note the similarity to the standard formula for the mean:

Problem: The deviations from the mean sum to zero Problem: The deviations from the mean sum to zero. (We proved this previously.) How far are scores, on average, from the mean? Zero (?!) Recall that the deviations sum to zero because the mean is a “balancing point” for a set of scores--the point at which the “weight” of the scores above counterbalances the “weight” of the scores below.

One solution: Average the absolute value of the deviation scores. Average Absolute Deviation: How far the typical (i.e., average) score is from the mean.

A second solution: Average the squared deviation scores Variance: The average squared deviation score.

A third solution: Take the square root of the average of the squared deviation scores Standard deviation: The square root of the average squared deviation score In our example, the square root of 7 is 2.65. The standard deviation is the square root of the variance. *** What does this tell us? It tells us how far people are from the mean, on average. (Ignoring whether people are above or below the mean.)

Relative scores Standard(ized) scores, or z scores 𝑧= 𝑥 − 𝑋 𝑠 𝑧= 𝑥 − 𝑋 𝑠 Note: s is the standard deviation. z indicates number of standard deviations the original score is above or below the original mean Result: M = 0, SD = 1 Try it yourself in Tutorial 3

Why are z-scores useful? Enables comparisons of different distributions (with different means, SDs) – similar idea repeated in Seminar 5 Imagine you have take two math test Test A: 0-30 possible marks Test B: 0-100 possible marks You score 25 for Test A and 85 for Test B. Did you do equally well – in comparison to your classmates?

Summary We have three measures: Average Absolute Difference Variance Standard deviation The variance and standard deviation are used most frequently. Why? Mathematically, it is easier to work with squared functions than absolute value functions.