Central Tendency

Chapter 3 (part 2)

The weighted mean

When you want to combine two sets of scores and get the overall mean.

You cannot just add the two means because they are relative (influenced by) to the sample size. So you use the weighted mean.

Weighted MeanOverall mean = M = ∑X (Overall sum for the combined group) =

∑X1 + ∑X2 n1+n2

Group-1 scores: 3, 4, 5, 6 Group-2 scores: 1, 4

Group-1 mean = ∑X1 = 18 = 4.5 n1 4

Group-2 mean = ∑X2 = 5 = 2.5 N2 2

Overall mean = ∑X1 + ∑X2 = 18 + 5 = 23 = 3.83 (not the same as n1+n2 4 + 2 6 4.5+2.5)

Characteristics of the mean

• Changing a score– Changing a single score in the sample produces a

new man• Introducing a new score or removing a score

– Whenever the N or ΣX changes mean changes• Adding or subtracting a constant form each

score– The same constant will be added or subtracted from

the mean• Multiplying or dividing each score by a constant

– Leads to the same transformation of the mean

The Median

• When a data set is ordered, it is called a data array.

• The median is defined to be the midpoint of the data array.

• The symbol used to denote the median is MD.

The median divides the area in the graph exactly in half.

The Median - Example

• The weights (in pounds) of seven army recruits are 180, 201, 220, 191, 219, 209, and 186. Find the median.

• Arrange the data in order and select the middle point.

The Median - Example

• Data array: 180, 186, 191, 201, 209, 219, 220.

• The median, MD = 201.

The Median

• In the previous example, there was an odd number of values in the data set. In this case it is easy to select the middle number in the data array.

The Median

• When there is an even number of values in the data set, the median is obtained by taking the average of the two middle numbers.

The Median - Example • Six customers purchased the following

number of magazines: 1, 7, 3, 2, 3, 4. Find the median.

• Arrange the data in order and compute the middle point.

• Data array: 1, 2, 3, 3, 4, 7.

• The median, MD = (3 + 3)/2 = 3.

Figure 3-6 (p. 64)A population of N = 6 scores with a mean of = 4. Notice that the mean does not necessarily divide the

scores into two equal groups. In this example, 5 out of the 6 scores have values less than the mean.

Difference between the Mean and the Median