Each of these three measures has its own properties. Most of the time we use these properties as basis for determining what measure to use to represent the center of the distribution.
As mentioned before the mean is the most commonly used measure of central tendency since it could be likened to a “center of gravity” since if the values in an array were to be put on a beam balance, the mean acts as the balancing point where smaller observations will “balance” the larger ones as seen in the following illustration.
Note that the frequency represented by the size of the rectangle serves as ‘weights’
in this beam balance.
To illustrate further this property, we could ask the student to subtract the value of the mean to each observation (denoted as di) and then sum all the differences. The computation can also be done alternatively as shown in the following table.
Monthly and positive deviations cancel out and the sum is equal to zero.
12,000% 20,000%
% 24,000% 25,000% 32,250% 36,000% 40,000% 60,000%
In the expression given above, we could see that each observation has a contribution to the value of the mean. All the data contribute equally in its calculation. That is, the “weight” of each of the data items in the array is the reciprocal of the total number of observations in the data set, i.e. 1 !.
Means are also amenable to further computation, that is, you can combine subgroup means to come up with the mean for all observations. For example, if there are 3 groups with means equal to 10, 5 and 7 computed from 5, 15, and 10 observations respectively, one can compute the mean for all 30 observations as follows:
! = !!!!+!!!!!+ !!!!!
30 = 10×5! + 5×15! + 7×10!
30 = 195 30 = 6.59
If there are extreme large values, the mean will tend to be ‘pulled upward’, while if there are extreme small values, the mean will tend to be ‘pulled downward’. The extreme low or high values are referred to as ‘outliers’.’Thus, outliers do affect the value of the mean.
To illustrate this property, we could tell the students that if in case there is one family with very high income of 600,000 pesos monthly instead of 60,000 pesos only, the computed value of mean will be pulled upward, that is,
! = 12,000 + 12,000 + ⋯ + 600,000 35 = 2,130,250
35 = 60,864.29
Thus, in the presence of extreme values or outliers, the mean is not a good measure of the center. An alternative measure is the median. The mean is also computed only for quantitative variables that are measured at least in the interval scale.
Like the mean, the median is computed for quantitative variables. But the median can be computed for variables measured in at least in the ordinal scale. Another property of the median is that it is not easily affected by extreme values or outliers.
As in the example above with 600,000 family monthly income measured in pesos as extreme value, the median remains to same which is equal to 32,250 pesos.
For variables in the ordinal, the median should be used in determining the center of the distribution. On the other hand, the mode is usually computed for the data set which are mainly measured in the nominal scale of measurement. It is also sometimes referred to as the nominal average. In a given data set, the mode can easily be picked out by ocular inspection, especially if the data are not too many. In some data sets, the mode may not be unique. The data set is said to be unimodal
if there is a unique mode, bimodal if there are two modes, and multimodal if there are more than two modes. For continuous data, the mode is not very useful since here, measurements (to the most precise significant digit) would theoretically occur only once.
The mode is a more helpful measure for discrete and qualitative data with numeric codes than for other types of data. In fact, in the case of qualitative data with numeric codes, the mean and median are not meaningful.
The following diagram provides a guide in choosing the most appropriate measure of central tendency to use in order to pinpoint or locate the center or the middle of the distribution of the data set. Such measure, being the center of the distribution
‘typically’ represents the data set as a whole. Thus, it is very crucial to use the appropriate measure of central tendency.
KEY POINTS
• A measure of central tendency is a location measure that pinpoints the center or middle value.
• The three common measures of central tendency are the mean, median and mode.
• Each measure has its own properties that serve as basis in determining when to use it appropriately.
ASSESSMENT
Note: Answers are provided inside the parentheses and italicized.
1. Thirty people were asked the question, “How many people do you consider your best friend?” The graph below shows their responses.
What measure of central tendency would you use to find the center for the number of best friends people have? Explain your answer. (Since there is a presence of an outlier, one can use the median which is numerically equal to 3) 2. The mean age of 10 full time guidance counselors is 35 years old. Two new full
time guidance counselors, aged 28 and 30, are hired. Five years from now, what would be the average age of these twelve guidance counselors? (The sum of ages is 350 for 10 counselors, with the two newly hired, the sum is now 408, thus yielding a mean currently at 34 years. Five years from now, the mean will go up to 39 years for the 12 guidance counselors.)
3. Houses in a certain area in a big city have a mean price of PhP4,000,000 but a median price is only PhP2,500,000. How might you explain this best? (There is an outlier (an extremely expensive house) in the prices of the houses.)
4. Five persons were asked on the usual number of hours they spent watching television in a week. Their responses are: 5, 7, 3, 38, and 7 hours.
a. Obtain the mean, median and mode. (The mean is 12; median is 7, mode is 7.)
b. If another person were to be asked the same question and he/she responded 200 hours, how would this affect the mean, median and mode? (Median and mode unchanged; mean increases to 43.3)
0%
5. For the senior high school dance, there is a debate going on among students regarding the color that will be featured prominently. Votes were sent by students via SMS, and the results are as follows:
Color Red Green Orange White Yellow Blue Brown Purple No. of
Votes
Received 300 550 70 130 220 710 35 5
a. Is there a clear winner on the choice of color? (Yes)
b. Compute for the mean, median and modal color (if possible). (We cannot compute for the mean and median. But the modal color is said to be blue.) c. Why is it that we could or could not find each measure of the central
tendency? (We cannot compute for the mean and median since color is a qualitative variable and is measured at the nominal level)
d. Which measure of central tendency will determine the color to be prominently used during the senior high school dance? (mode)
6. Everyone studied very hard for the quiz in the Statistics and Probability Course.
There were 10 questions in the quiz, and the scores are distributed as follows:
Score Number of Students
a. Compute for the mean, median, and mode for this set of data. (The computation could be done as follows:
Score
7 5 35 14
6 3 18 9
5 2 10 6
4 0 0 4
3 1 3 4
2 1 2 3
1 0 0 2
0 2 0 2
Sum = 40 Sum = 304
Mean = µ = !"#!" = 7.6;
Median is the average of the 20th and 21st observations = !!!! = 8.5. Note that the 20th observation is 8 while the 21st observation is 9 based on the less than cumulative frequency.
Mode = 9 since that is the score with the highest frequency equal to 12.
c. Suppose the teacher said “Everyone in the class will be getting either the mean, median, or mode for their official score.”
i. What would students want to receive (mean, median, or mode)? (Mode) ii. Which would students want to receive the least (mean, median or mode)?
(Mean)
iii What is the fairest score to receive would be? Ask students to explain their answers. (Note: There is no right or wrong answer for this question. It all depends on the reasoning of the students)