How to Compute and Interpret the Mean, …
How to Compute and Interpret Measures of Central Tendency (Mean, Median, and Mode)
Objective: Learn how to compute and interpret and when to use measures of central tendency including the mean, median, and mode
Keywords and Concepts
1. Mean
2. Median
3. Mode
4. Central tendency
5. Midpoint or middle of data
6. [pic]
7. 50th percentile
8. Most frequently occurring score
9. Best measure of central tendency
10. Skewed data
The mean, median and mode, common measures of central tendency, represent either a typical or representative score and/or a value about which the data tend to center.
A measure of central tendency frequently lies at the midpoint or middle of a data set.
Mean
The arithmetic average (termed the mean and abbreviated as [pic]) represents the most appropriate measure of central tendency for continuous-type data. It is obtained by adding all of the scores and dividing this sum by the number of scores
Mean ([pic] ) = [pic] (eq. 1)
where the mean can be denoted by [pic] (pronounced ”X-bar”) for samples and µ (pronounced “mu”) when dealing with populations; ∑ denotes summation of a set of values; X represents the individual raw scores, and N equals the number of scores.
Example Mean Calculations
Compute the mean of the following scores placed in ascending order: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10:
Step 1. Sum up the scores.
∑X = 1+2+3+4+5+6+7+8+9+10 = 55
Step 2. Divide ∑X by the total number of scores or data points (in this example 10).
[pic] = 55 ÷ 10 (eq. 2)
= 5.5
Median
The median of a set of scores represents the middle value (50th percentile) when the scores are arranged as an array in order of increasing (or decreasing) magnitude. The median is often denoted by [pic] (pronounced “X-tilde”). The median often becomes a more appropriate (representative) measure of central tendency when the data are skewed— that is, the majority of scores tend to accumulate toward either the high or low end of the distribution with a few extreme scores at the opposite end.
To locate the median, first rank the scores and follow these two guidelines:
1. For an odd number of scores, the median is the middle score.
2. For an even number of scores, the median is the mean (arithmetic average) of the two middle scores.
Example Median Calculations
Example #1
Compute the median for these five scores: 10, 30, 27, 29, 12.
Step 1. Arrange the scores in ascending order.
10, 12, 27, 29, 30
Step 2. Because the number of total scores equals 5, an odd number, the third number (27) is the raw score at the exact middle and becomes the median [pic].
Example #2
Compute the median for these six scores: 5, 6, 8, 50, 10, 70.
Step 1. Arrange the scores in ascending order
5, 6, 8, 10, 50, 70
Step 2. Because the number of scores equals 6, an even number; compute the average of the values for the third and fourth “middle” scores. The arithmetic average of these raw scores of 8 and 10 equals the [pic], a score of 9.
Mode
The mode (denoted by M) represents the most frequently occurring score. When two scores occur with the same greatest frequency, each one equals the mode and the data set is considered bimodal. When more than two scores occur with the greatest frequency, the data set is said to be multimodal.
To determine the mode, locate the most frequently appearing number.
Example Calculations
Compute the mode of the following ten scores: 10, 29, 26, 28, 15, 10, 25, 27, 10, 29:
The mode, M, equals 10
Round-off Rule
Carry one more decimal place than present in the original set of numbers. Round-off only the final answer, not intermediate values. For example, the mean of the numbers 2, 3, and 5 equals 3.33333,which is then rounded to 3.3. [Because the original data were whole numbers, the answer rounds to the nearest tenth.] The mean of the numbers 2.1, 3.4, and 5.7 rounds to 3.73 (one more decimal place than used for the original values).
The Best Central Tendency Measure
There is no single “best” measure of central tendency. The different measures of central tendency have different advantages and disadvantages as summarized in Table 1.
|Table 1. Unique aspects of mean, median, and mode. |
|Average |Definition |How common |Existence |Takes every score |Affected by |Advantages/ |
| | | | |into account? |extreme scores? |Disadvantages |
|Mean |[pic]=∑X÷N |Most familiar average|Always exists |Yes |Yes |Takes every score into|
| | | | | | |account/ affected by |
| | | | | | |extreme scores |
|Median |Middle score |Commonly used |Always exists |No |No |Good choice if there |
| | | | | | |are extreme scores |
|Mode |Most frequent score |Rarely used |Always exists |No |No |Appropriate for |
| | | | | | |nominal data |
|Comments: For a symmetrical distribution of data with one mode, the mean, median, and mode are about the same value. |
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“The Average Male”
The “average” American male is named “Robert”; he is 31 years old, 5 ft 9 1/2 in tall, weighs 172 pounds; wears a size 40 suit, and 9 1/2 inch shoes; he has a 34 inch waist. Each year “Robert” will eat 12.3 pounds of pasta, 26 pounds of bananas, 4 pounds of potato chips, 18 pounds of ice cream, and 79 pounds of beef; he will watch 2567 hours of television per year and get 585 pieces of mail; he will get 7.7 hours of sleep per night and commute 21 minutes to work, at which he will work for 6.1 hours. Robert will marry 1.4 times and have 2.1 children.
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