Following are the number of nonsense syllables (out of 20) correctly recalled after rehearsal by 73 sample participants. Calculate the mean.
X f fX
17 1 –––––––
16 0 –––––––
15 3 –––––––
14 2 –––––––
13 4 –––––––
12 8 –––––––
11 12 –––––––
10 13 –––––––
9 6 –––––––
8 8 –––––––
7 5 –––––––
For Example
Let us calculate the mean for the following scores from a sample arranged into a simple fre- quency distribution table:
X f fX X f fX
48 1 48 41 4 164
47 4 188 40 6 240
46 2 92 39 3 117
45 4 180 38 0 0
44 9 396 37 1 37
43 8 344 36 2 72
42 5 210 35 1 35
n = 50 ∑fx = 2123 M fX
n 2123 50
42 46.
X f fX
6 3
5 4
4 2
3 1
2 0
1 1 –––––––
M fX
�n M = _________
Your Turn!
(continued)
Mean for a Grouped Frequency Distribution
The procedure for calculating the mean for scores that have been arranged into a grouped frequency distribution is the same as for a simple frequency distribution except that you use the midpoint of each class interval for X .
For Example
Let us calculate the mean for the following scores from a sample arranged into a grouped frequency distribution:
Class interval Midpoint ( X ) f fX
36–38 37 4 148
33–35 34 3 102
30–32 31 1 31
27–29 28 4 112
24–26 25 7 175
21–23 22 6 132
18– 20 19 6 114
15–17 16 2 32
12–14 13 0 0
9–11 10 3 30
n = 36 ΣfX = 876
M fX
n 876 36
24 33.
Advantages of the Mean
• The mean is the measure of central tendency that is used most often in statistics because it is the only one of the three that takes every score value into consideration. It is there- fore the only one that can be used for additional statistical operations.
• The sample mean (M) is also an unbiased estimate of the population mean (à) and is therefore important in inferential statistics.
Mean (M) as an Unbiased Estimate of à
Here is an explanation of an unbiased estimate of the population mean. Remember that with inferential statistics, we use sample statistics to estimate population parameters.
Suppose we were to select a random sample of people from a population, measure them on some dependent variable, and then calculate a mean. If we used that sample mean to esti- mate the population mean, our estimate would probably not be exactly equal to the popula- tion mean. It might either underestimate or overestimate the population mean. This would be the sampling error referred to in Chapter 1.
If we drew another sample from the same population and again calculated a mean, it would again probably be somewhat different from the actual population mean. And if we repeated this process over and over, our sample means would likely continue to be somewhat off. However, there would be no systematic pattern in our deviations from the population mean. Errors in both directions (overestimates and underestimates) would balance out. Thus, there would be no systematic bias in our sample estimates of the population mean, even though any particular estimate might be somewhat off due to sampling error.
In later chapters, we will learn ways to measure the amount of sampling error to be expected when estimating population parameters.
WHEN TO USE WHICH MEASURE OF CENTRAL TENDENCY
In determining which measure of central tendency to use for a set of scores, the scale of mea- surement and the shape of the distribution need to be considered.
Scale of Measurement
• While the mode can be used for all scales (nominal, ordinal, interval, ratio), it is the only measure of central tendency that can be used for nominal variables, and that is its typical use.
• The median can be used for all scales except nominal. (It makes no sense to determine the median for categorical variables such as the median color of automobiles.)
• The mean can be used only for variables measured on interval or ratio scales.
Shape
• For approximately normally shaped distributions, the mean is preferred.
• For skewed distributions, use the median. This is because the mean is affected by extreme scores and the median is not. Such extreme scores are referred to as outliers.
Given the following data set, the mean is 74.11:
36, 54, 56, 70, 72, 91, 91, 95, 102
However, if the last score of 102 were replaced with 802, the mean would be affected by that extreme score and would be pulled in that direction to result in a value of 151.89. On the other hand, the median would not be affected by the outlier and in both cases would result in a value of 72, which more accurately reflects the central tendency of the entire distribution.
THE POSITION OF CENTRAL TENDENCIES IN FREQUENCY POLYGONS
In a normal distribution, all three measures of central tendency would be in the middle of the distribution and all would have the same value. The mode will always be at the peak, reflecting the highest frequency. The median divides the distribution in half, and the mean balances out the high and low scores.
In skewed distributions, the mode would again be at the peak; the mean would be located toward the tails in the direction of the skew (having been affected by either high or low extreme scores); and the median would be between the mode and the mean (so that half the scores lie above it and half below it).
Answers to “Your Turn!” Problems I. Simple Frequency Distribution and Mode
A. X f
17 1
16 5
15 9
14 4
13 3
12 2
MO = 15 B. MO = 72
II. Median: Counting Method
A. 78, 73, 70, 67, 66, 65, 65, 64, 62, 60, 56 11 1
2 6 Sixth score is 65, thus Mdn = 65.
Your Turn!