Shepherdville College: STATISTICS: Measure of Central Tendency
Shepherdville College: STATISTICS: Measure of Central Tendency
Shepherdville College: STATISTICS: Measure of Central Tendency
Learning objectives:
a. Analyze mean, median, mode and weighted mean to establish a
conjecture;
b. Evaluate problems involving mean, median, mode and weighted
mean.
c. Create one’s methods and approaches to easily determine the
mean, median, mode and weighted mean.
Definition of Terms:
1. Mean – is the central value of a discrete set of numbers: specifically,
the sum of the values divided by the number of values.
2. Median – is the middle number in a sorted, ascending or descending,
list of numbers and can be more descriptive of that data set than the
average.
3. Mode – is the number that appears most frequently in a data set..
4. Weighted Mean – A mean where some values contribute more than others.
When the weights add to 1: just multiply each weight by the matching
value and sum it all up.
Σx
mean=
n
Statisticians often collect data from small portion of a large group in
order to determine information about the group. In such situation the entire
group under consideration is known as the population, and any subset of the
population is called a sample. It is traditional to denote the mean of a
sample by x
̅ (which is read as “x bar”) and to denote the mean of a population
by the Greek letter μ (lowercase mu).
Example 1
Six friends in a biology class of 20 students received test
grades of
92, 84, 65, 76, 88, and 90
Find the mean of these test scores.
Solution
The 6 friends are a sample of the population of 20 students. Use
x
̅ to represent the mean.
Example 2
Find the median of the data in the following lists.
a. 4, 8, 1, 14, 9, 21, 12
b. 46, 23, 92, 89, 77, 108
Solution
a. The list 4, 8, 1, 14, 9, 21, 12 contains 7 numbers. The median of
a list with an odd number of entries is found by ranking the
numbers and finding the middle number. Ranking the numbers from
smallest to largest gives
1, 4, 8, 9, 14, 12, 21
The middle number is 9. Thus 9 is the median.
b. The list 46, 23, 92, 89, 77, 108 contains 6 numbers. The median
of a list of data with an even number of entries is found by
ranking the number and computing the mean of the two middle
numbers. Ranking the numbers from smallest to largest gives
23, 46, 77, 89, 92, 108
The two middle number s are 77 and 89. The mean of 77 and 89 is
83. Thus 83 is the median of the data.
The mode
The mode of a list of numbers is the number that occurs most
frequently.
Some list of numbers do not have a mode. For instance, in the list 1,
6, 8, 10, 32, 15, 49, each numbers occurs exactly once. Because no number
occurs more often than the other numbers, three in no mode.
Example 3
Find the mode of the data in the following lists.
Solution
a. In the list 18, 15, 21, 16, 15, 14, 15, 21, the number 15 occurs
more than the other numbers. Thus 15 is the mode.
Σx 506,000
mean= = =101,200
n 5
Answer: The median will remain the same because 11 will still be the middle
number in the ranked list.
The median is the middle number, $36,000. Because the $20,000 salary
occurs the most, the mode is $20,000. The data contain one extreme value that
is much larger than the other values. This extreme values makes the mean
considerably larger than the median. Most of the employees of this company
would probably agree that the median of $36,000 better represents the average
of the salaries than does either the mean or he mode.
Σ( x ∙ w)
Weighted mean=
Σw
Where Σ( x ∙ w) is the sum of the products formed by multiplying each
number by its assigned weight, and Σ w is the sum of all weights.
Table above shows Dillon’s fall semester course grades. Use the
weighted mean formula to find Dillon’s GPA for the fall semester.
Solution
The B is worth 3 points, with a weight of 4; the A is worth 4-
points with a weight of 3; the D is worth 1 point, with a weight of 3;
and the C is worth 2 points, with a weight of 4. The sum of all the
weights is 4 + 3 + 3 + 4, or 14.
( 3 × 4 ) + ( 4 ×3 )+ (1 ×3 )+(2 × 4)
Weighted mean=
14
35
¿ =2.5
14
2 0 3 1 2 1 0 4
2 1 1 7 2 0 1 1
0 2 2 1 3 2 2 1
1 4 2 5 2 3 1 2
2 1 2 1 5 0 2 5
Number of laptop Computers per Household
The formula for weighted mean can be used to find the mean of the data
in a frequency distribution. The only change is that the weights w1, w2, w3,
…, wn are replaced with the frequency f1, f2, f3, …, fn. this procedure is
illustrated in the next example.
Example 5
Find the mean of the data in table *
Solution
The numbers in the right-hand column of table * are the
frequencies f for the numbers in the first column. The sum of all the
frequencies is 40.
3. Quiz Scores
Scores on biology Frequency
quiz
2 1
4 2
6 7
7 12