Unit E: Statistics and Probability

Chapter 1: Statistics

Mean


Measures of central tendency are numbers that describe what the average is within a set of data. There are three main measures of central tendency: mean, median, and mode. Although these measures are calculated differently, each represents an average for the set of data.

The most common measure of central tendency is the mean. The mean is represented by the symbol   x ¯ .

To calculate the mean, x ¯ ,
  1. add all the values
  2. divide by the number of values

x ¯ = sum   of   the   values   in   the   data   set total   number   of   values   in   the   data   set = x 1 + x 2 + x 3 + … + x n n
where
x1 is the first value in the data set
x2 is the second value in the data set
x3 is the third value in the data set
xn is the last value in the data set
n is the number of values in the data set


Note: The value of n is the numerator and in the denominator will always be the same.

x ¯ = sum   of   the   values   in   the   data   set total   number   of   values   in   the   data   set = x 1 + x 2 + x 3 + … + x n n

Eleven starting salaries for a certain company are listed in the table. Calculate the mean starting salary for new employees.

New Employee Starting Salary
(in thousands of dollars)
$31
$35
$35
$39
$44
$45
$49
$61
$73
$85
$97
 


x ¯ = sum   of   the   values   in   the   data   set total   number   of   values   in   the   data   set = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 + x 11 n = 31 + 35 + 35 + 39 + 44 + 45 + 49 + 61 + 73 + 85 + 97 11 = 594 11 = 54

The mean starting salary at the company is $54 000.

The mean formula is also useful to find one of the values in a set of data if the mean and all the other values in the set of data are known.

Jared received the following marks on his math tests: 87%, 95%, 76%, 88%. What is the minimum mark that he must earn on the last test in order to obtain an average of at least 85%?

Let x = the unknown value, which is the minimum mark that Jared must earn to obtain an average of 85%.

x ¯ = sum   of   the   values   in   the   data   set total   number   of   values   in   the   data   set x ¯ = x 1 + x 2 + x 3 + x 4 + x 5 n
 
Substitute 85% for the mean that Jared is aiming to earn and y, the minimum mark required, into the mean formula.



85 = 87 + 95 + 76 + 88 + x 5 85 = x + 346 5
 
Multiply both sides by 5 to remove the fraction.



85 × 5 = x + 346 5 × 5 85 × 5 = x + 346 5 × 5 425 = x + 346
 
Subtract 346 from both sides to isolate x.



425 - 346 = x + 346 - 346 79 = x

Jared needs a minimum of 79% on his next test to earn an average of at least 85%.