Central Tendency

The measure of central tendency is a single number that represents an entire list of numbers when analyzing data.    One such measure is called the average or mean.

Mean

The mean or average is the sum of values divided by the total number of values.

Example 1:

Find the average or mean of the following:   67, 98, 32, 45, 88, 78.

 

Solution:

The sum of the 6 numbers is 68.

Therefore the mean is 68.





Student #1:

Find the mean of:
23, 65, 45, 89, 3, 5, 45, 21

37               45               62               3               55              

Answer

Example 2:

Betty scores an 86, 75 and 90 on the first three tests.   What will she need on the fourth test in order to have an average test score of 80?

 

Solution:

The average or mean is the sum of the scores divided by 4.   Let x represent the missing fourth test.

  

Solve for x by multiplying both sides by 4.

251 + x = 320

Subtract 251 from both sides.

251 - 251 + x = 320 - 251

x = 69


Therefore Betty will need a score of 69 on the fourth test to have an average of 80.

Student #2:

Ben scores a 76, 78 and 89 on the first three tests.   What will he need on the fourth test in order to have an average test score of 70?

71               76               37               55               59              

Answer

Weighted Mean or Weighted Average

When analyzing data, some numbers may appear more than one time.   If this pattern is present, then the mean can then be calculated using the property of multiplication.

Example 3:

Find the weighted mean of   3,3,3,3,5,5,7,8,8,8,8,8,9,9,9,10,10,12,13,13.

Solution:
Write the sum of the numbers in terms of multiplication.

Therefore the weighted mean is   7.7.

This problem can also be calculated using a table format.

Find the weighted mean using the table.

Value	Frequency
3	4
5	2
7	1
8	5
9	3
10	2
12	1
13	2

Solution:

The weighted mean is 7.7.

Student #3:

Find the grade point average for the following student.
(The g.p.a. is the same as the weighted mean.)

Class	Units	Grade
Math 124	3	A
English 101	4	C
Biology 101	3	B
History 3a	2	A
P.E. 5a	2	D

 

1.87               2.4               3.7               2.8               3.4              

Answer

Median

The median is another way to view central tendency.   If the list of numbers is odd, the median is the value of the middle number.   If the list of numbers is even, then the median is the mean of the two middle valued numbers.

Example 4:

Find the median of:

8, 43, 76, 34, 23

 

Solution:

Arrange the numbers in numerical order.

8, 23, 34, 43, 76

Since the list has an odd number of elements, the median is the middle number.

Therefore the median is 34.

________________________________________________

Example 5:

Find the median of:

28, 54, 63, 24, 33, 87

 

Solution:

Arrange the numbers in numerical order.

24, 28, 33, 54, 63, 87

Since the list has an even number of elements, the median is the mean of the two middle numbers.

Therefore the median is 43.5.

Student #4:

Find the median of the following list.
76, 34, 22, 92, 45, 28

40               38               39.5               34               45              

Answer

Mode

The mode is the number in the list that occurs most often.

Example 6:

Find the mode   51, 76, 65, 51, 87, 99.

 

Solution:

The number 51 occurs the most often in the list.

Therefore 51 is the mode.

________________________________________________

Example 7:

Find the mode   34, 56, 87, 34, 87, 66, 54, 22.

 

Solution:

Since both 34 and 87 appear twice in the list, the list is bimodal.


Therefore the modes are 34 and 87.

A list may also have no mode.

Student #5:

Find the mode of:
67, 87, 66, 87, 92, 34

34               92               66               67               87              

Answer

Homework


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Student #1:    The answer is A:


Solution:

The sum of 23, 65, 45, 89, 3, 5, 45, 21 is 296.   The quotient of 296 and 8 is 37.


Therefore the mean is 37.

Go Back to Lesson

Student #2:    The answer is C:

Solution:
The average or mean is the sum of the scores divided by 4.   Let x represent the missing fourth test.

Solve for x by multiplying both sides by 4.



x + 243 = 280

Subtract 243 from both sides.

x + 243 - 243 = 280 - 243

x = 37


Therefore the fourth test must be a 37.

Go Back to Lesson

Student #3:    The answer is D:

Solution:

The weighted mean is as follows:

The grade of an A is weighted as 4, a B is weighted as 3, a C is weighted as 2, and a D is weighted as 1.





Therefore the grade point average is   2.8.

Go Back to Lesson

   Student #4: The answer is C:

Solution:

Find the median of the following list.

76, 34, 22, 92, 45, 28

Arrange the numbers in numerical order.

22, 28, 34, 45, 76, 92

Since the list has an even number of elements, the median is the mean of the two middle numbers.

The sum of 34 and 45 is 79.

The quotient of 79 and 2 is 39.5


Therefore the median is 39.5.

Go Back to Lesson

Student #5:    The answer is E:


Solution:

Find the mode of:
67, 87, 66, 87, 92, 34

The number 87 occurs the most often in the list.


Therefore the mode is 87.

Go Back to Lesson