Download How to Compute and Interpret the Mean, Median, and Mode

How to Compute and Interpret the Mean, Median, and Mode
MVS 250 – V. Katch
Page 1
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
6. X
2. Median
7. 50th percentile
3. Mode
8. Most frequently occurring score
4. Central tendency
9. Best measure of central tendency
5. Midpoint or middle of data
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
X ) 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 ( X ) =
X
N
(eq. 1)
where the mean can be denoted by X (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.
How to Compute and Interpret the Mean, Median, and Mode
MVS 250 – V. Katch
Page 2
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).
X
= 55 ÷ 10
N
(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 X˜ (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
How to Compute and Interpret the Mean, Median, and Mode
MVS 250 – V. Katch
Page 3
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 X˜ .
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 X˜ , 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:
How to Compute and Interpret the Mean, Median, and Mode
MVS 250 – V. Katch
Page 4
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
Takes every
score into
account?
Affected
by extreme
scores?
Definition
How common
Existence
Mean
X =∑X÷N
Most familiar
average
Always
exists
Yes
Yes
Median
Middle score
Commonly
used
Always
exists
No
No
Mode
Most
frequent
score
Rarely used
Always
exists
No
No
Advantages/
Disadvantages
Takes every
score into
account/
affected by
extreme scores
Good choice if
there are
extreme scores
Appropriate
for nominal
data
How to Compute and Interpret the Mean, Median, and Mode
MVS 250 – V. Katch
Page 5
Comments: For a symmetrical distribution of data with one mode, the mean, median, and mode are
about the same value.
“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.