Today’s Questions

• How can we summarize a distribution of scores efficiently using quantitative (as opposed to graphical) methods?

Measures of Central Tendency

• Central tendency: most “typical” or common score

(a) Mode

(b) Median

(c) Mean

Measures of Central Tendency

1. Mode: most frequently occurring score

10, 20, 30, 40, 40, 50, 60

Mode = 40

Measures of Central Tendency

2. Median: the value at which 1/2 of the ordered scores fall above and 1/2 of the scores fall below

1 2 3 4 5 1 2 3 4

Median = 3 Median = 2.5

Measures of Central Tendency

x = an individual score

N = the number of scores

Sigma or = take the sum

• Note: Equivalent to saying “sum all the scores and divide that sum by the total number of scores”

xN

MX1__

3. Mean: The “balancing point” of a set of scores; the average

A

B

C D E

3 4 5 6 7 8 9

(-1)

(-2)

(-2) (+4)

(+1)

(– 1) + (– 2) + (– 2) + 1 + 4 = 0

A

B

C D E

3 4 5 6 7 8 9

(+2)

(-1)

(-3)

(- 4)

(- 4)

(– 1) + (– 3) + (– 4) + (– 4) + 2 = –10

0Mx We begin by noting that, when we have found a proper balancing point, the sum of all the mean deviations is 0.00.

What we want to do next is solve this equation for M.

0Mx We first distribute the summation operation and

Mx move one term to the right-hand side.

MNx Next, we note that the sum of a bunch of M’s is simply the number of M’s (N) times M.

MN

x 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

When the distribution of scores is normal, the mode = median = mean

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

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