Mean, Median and Mode of a Data Set

Examples on how to find the mean, median and mode of a data set of real numbers are presented along with their detailed solutions and explanations. The mean, median and mode are statistical measures of central tendencies that in a well distributed data tends to summarize a whole data set with a single value. These three measures are easy to understand and to use in comparing data sets.
The outliers are also defined and discussed and their effects on the mean, median and mode discussed through examples with their solutions.
Online calculators to compute the Mean , the Median and the Mode are included.

Mean of a Data Set

The arithmetic mean (or average) of a data set is defined as the sum of all data values in the set divided by the number of values in this set. If x1, x2, x3 ... xN are the values in a data set, the mean μ is given by the formula
 
\mu = \dfrac{\sum_{i=1}^{N} x_i}{N}

Example 1
In a math exam, the students in class A scored 55, 72, 96, 92, 87, 52, 92, 45, 58, 77, 86, 80 and the students in class B scored 55, 67, 75 , 95, 82, 86, 38, 90, 42, 56, 82, 96. What is the mean (average) score of each class? Use the two means to compare the performance of the two classes.
Solution to Example 1
There are 12 scores in class A hence N = 12. Use the formula to calculate the mean for class A as follows
 
\mu_A = \dfrac{55 + 72 + 96 + 92 + 87 + 52 + 92 + 45 + 58 + 77 + 86 + 80}{12} = 74.3
There are 12 scores in class B hence N = 12. Use the formula to calculate the mean for class B as follows
 
\mu_B = \dfrac{55 + 67 + 75 + 95 + 82 + 86 + 38 + 90 + 42 + 56 + 82 + 96}{12} = 72
The mean (average) of class A is higher that the mean of class B. On average, class A "scored" better than class B, but If we examine the two sets of scores, not all students in class A scored higher than all students in class B.
For any given data set of values, the mean is larger that the smallest value and smaller that the largest value. In class A above the smallest score is 55 and the largest score is 96 and the mean is 74.3. In class B, the mean is 72 and the smallest and largest scores are 55 and 96 respectively.

Median of a Data Set

The median of a data set is the middle data value after ordering the given data. If the number N of data values is odd, the median is the middle data vale and if N is even, the median is the average of the two data values in the middle.
Example 2
The heights, in centimeters, of a group of 11 children of different ages are follows: 110, 105, 126, 65, 134, 102, 78, 80, 119, 67, 88. What is the median height of this group of children?
Solution to Example 2
We first order the given data from smallest to the largest heights.
65 , 67 , 78 , 80 , 88 ,
102 , 105 , 110 , 119 , 126 , 134
There is a total of 11 data values and the one in the middle is the median and is equal to 102. There are close to 50% of the data values that are lower than the median (102) and close to 50% of data values higher than the median.


Example 3
The weights, in kilograms, of a group of 12 men are follows: 110, 82, 99, 70, 77, 87, 78, 80, 102, 79, 88, 95. What is the median weight of this group of men?
Solution to Example 3
We first order the given data from smallest to the largest weights.
70 , 77 , 78 , 79 , 80 ,
82 , 87 , 88 , 95 , 99 , 102 , 110 ,
There is a total of 12 data values and the average of the two data values 82 and 87 in the middle is the median and is equal to (82 + 87) / 2 = 84.5. Exactly 50% of the data values are lower that the median (84.5) and 50% of data values are higher than the median.

Mode of a Data Set

The mode of a data set is the data value that occurs with the highest frequency.
Example 4
In an exam, students in a class scored as follows: 45, 67, 95, 89, 88, 40, 90, 88, 56, 78, 88, 76. What is the mode of the given scores?
Solution to Example 4
We first order given scores from smallest to the largest weights.
40 , 45 , 56 , 67 , 76 , 78 ,
88 , 88 , 88 , 89 , 90 , 95
We then look for the data value that occurs the most which in this example is 88.


Example 5
A group of people were asked about the number of brothers and sisters they have and they answered as follows: 2, 3, 3, 4, 4, 3, 2, 3, 4, 5, 2, 4, 2. What is the mode of this data set?
Solution to Example 5
We first order given data from smallest to the largest weights.
2 , 2 , 2 , 2 , 3 , 3 , 3 , 3 , 4 , 4 , 4 , 4 , 5
There are three data values that occur four times each and therefore have the same frequency of occurrence. This data set has 3 modes which are: 2, 3 and 4.

The Mean, Median and Mode are Measures of Central Tendency

The Mean, Median and Mode are single value quantities that tend to describe the center of a data set. For a data set where data values are close to each other, the three quantities tend to be close in value and describe the typical central data value.
Example 6
A student scored 89, 90, 92, 96,91, 93 and 92 in his math quizzes. Find the mean, median and mode of these scores.

Solution to Example 6

mean
\text{mean} = \dfrac{89 + 90 + 92 + 96 + 91 + 93 + 92}{7} = 91.9
median
Order the data from the smallest to the largest
89 , 90 , 91 ,
92 , 92 , 93 , 96
The median is the data value located in the middle and it is equal to 92.
Mode
89 , 90 , 91 ,
92 , 92 , 93 , 96
The mode is the data value that is most repeated and it is equal to 92.
For this data set, we can use any of the three central tendencies (mean, median or mode) to describe a typical central data value because thay are close in value. This is not always the case however. Data sets with outliers may have their central tendencies affected as we will examine examples below.

Outliers and their Effects on the Measures of Central Tendencies

Outliers are extremely low values or extremely high values in a data set. They may affect the mean, median or mode.
Example 7
A student scored 0, 90, 88, 96, 92, 88 and 95 in his math quizzes. Answer the following questions:
a) Which score may be considered as an outlier?
b) Calculate the mean, median and mode with and without the outlier and make a decision as to which central tendency better describes a typical central score.
Solution to Example 7
a) The lowest score of 0 may be considered as an outlier because it is much lower than the next higher score of 88.
b) with the outlier 0, the scores are: 0, 90, 88, 96, 92, 88 and 95
mean = 78.4 , median = 90 , mode = 88
Without the outlier 0, the scores are: 90, 88, 96, 92, 88 and 95
mean: = 91.5 , median = 91 , mode = 88
The outlier 0 tends to affect the mean as the difference between the means with (78.4) and without (91.5) outlier is relatively large and therefore the median and the mode would better describe a typical score of this student.

Data Dispersion and its Effects on the Measures of Central Tendencies

Data may be considered dispersed if the data values are far from the average value (mean) of the whole data set. Measures of central tendencies may not be enough or even sometimes suitable to describe a typical central data value.
Example 8
A class scored 96, 20, 20, 45, 40, 32, 97, 100, 98, 45, 90, 35 and 91 in an exam.
Calculate the mean, median and mode make a decision as to which measure of central tendency better describes the typical central score.
Solution to Example 8
mean = 62.2 , median = 45 , mode = 20
The scores are greatly dispersed with some scores much lower than the mean and other scores higher than the mean. This has greatly affetced the mode and to some extent the meadian. The mean, median and mode differ in values because the scores are dispersed with the smallest score of 20 and the largest score of 100. The mean may, in this situation, be used as a measure of central tendency but the mode of 20 is definitely not a typical central score.
The standard deviation is a measure of the dispersion of the data values around the mean.

More Links and References

Mean and Standard deviation.
Central tendency
Mean
Median
Mode
Standard Deviation.

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