|
Mode:
The most frequently occurring score. A
distribution of scores can be unimodal (one score occurred most frequently), bimodal (two
scores tied for most frequently occurring), or multimodal. In the table below the mode is
32. If there were also two scores with the
value of 60, we would have a bimodal distribution (32 and 60).
Median:
The point on a rank ordered list of scores below which 50% of the scores fall. It is especially useful as a measure of central
tendency when there are very extreme scores in the distribution, such as would be the case
if we had someone in the age distribution provided below who was 120. If the number of scores is odd, the median is the
score located in the position represented by (n+1)/2. In the table below the median is
located in the 4th position (7+1)/2 and would be reported as a median of 42. If the number of scores are even, the median is the
average of the two middle scores. As an example, if we dropped the last score (65) in the
above table, the median would be represented by the average of the 3rd (6/2) and 4th
score, or 37 (32+42)/2. Always remember to order the scores from low to high before
determining the median.
Variableè
|
Age |
ç
Also known as X |
|
|
|
24 |
|
|
|
|
32 |
ç
Mode |
|
|
|
32 |
|
|
|
|
42 |
ç
Median |
Xi |
çEach
score |
|
55 |
|
|
|
|
60 |
|
|
|
|
65 |
|
|
|
|
|
|
|
|
n= |
7 |
ç
Number of scores (or cases) |
|
|
|
310 |
ç
Sum of scores |
|
|
|
44.29 |
ç
Mean |
|
|
Mean:
The sum of the scores ( ) is divided by the number of scores (n) to compute an arithmetic
average of the scores in the distribution. The
mean is the most often used measure of central tendency.
It has two properties: 1) the sum of the deviations of the individual scores
(Xi) from the mean is zero, 2) the sum of squared deviations from the mean is smaller than
what can be obtained from any other value created to represent the central tendency of the
distribution. In the above table the mean age is 44.29 (310/7).
Weighted
Mean:
When two or more means are combined to develop an aggregate mean, the influence of each
mean must be weighted by the number of cases in its subgroup.
Example
Wrong
Method:
Correct
Method: 
Range: The difference between the highest and lowest score
(high-low). It describes the span of scores
but cannot be compared to distributions with a different number of observations. In the table below, the range is 41 (65-24).
Variance: The average of the squared deviations between the
individual scores and the mean. The larger the
variance the more variability there is among the scores.
When comparing two samples with the same unit of measurement (age), the
variances are comparable even though the sample sizes may be different. Generally, however, smaller samples have greater
variability among the scores than larger samples. The
sample variance for the data in the table below is 251.57.
The formula is almost the same for estimating population variance. See formula in Appendix.
Standard
deviation:
The square root of variance.
It provides a representation of the variation among scores that is directly comparable to
the raw scores. The sample standard deviation
in the following table is 15.86 years.
Variableè |
Age |
|
|
ç
 44.29 |
|
24 |
-20.29 |
411.68 |
|
|
32 |
-12.29 |
151.04 |
|
|
32 |
-12.29 |
151.04 |
ç
squared deviations |
|
42 |
-2.29 |
5.24 |
|
|
55 |
10.71 |
114.70 |
|
|
60 |
15.71 |
246.80 |
|
|
65 |
20.71 |
428.90 |
|
|
|
|
|
|
n= |
7 |
|
1509.43 |
ç |
|
|
|
251.57 |
ç
sample variance |
|
|
|
15.86 |
ç
sample standard deviation |
Software Output Example
|
