Mean (Average)
The mean, also called the average, is a measure of central tendency of a group of values. Unless otherwise specified, mean usually refers to the arithmetic mean, as opposed to the geometric mean or harmonic mean.
If the data we are averaging is represented by the measurements \( x_1, x_2, x_3, \ldots, \) the mean is written \( \bar x \) and pronounced "x bar." To calculate the mean, we take the sum of all values and divide by the number of values. Specifically,
\[ \bar x = \frac{x_1 + x_2 + x_3 + \cdots + x_n}{n}. \]
Note that if the mean is calculated by sampling from a larger population, it is called the sample mean, \( \overline{x}, \) to distinguish it from the population mean \((\)denoted \( \mu \) or \( \mu_x). \) The population mean is the arithmetic mean of the entire population, which is only approximated by the mean of a sample, although the law of large numbers proves that the sample mean approaches the population mean as the sample size grows.
Other measures of central tendency that might be used in place of the mean are median and mode.
What is the mean of the values \( \{ 1, 2, 2, 5, 5 \}?\)
There are a total of \(5\) values.
The mean is \( \frac{ 1 + 2 + 2 + 5 + 5 } { 5 } = \frac{ 15 } { 5} = 3 \). \(_\square\)
Use the concept of mean to solve the following problem:
I am thinking of \(4\) numbers.
The average of the \(4\) numbers is \(44.\)
The average of the first \(3\) numbers is \(33.\)
What is the value of the \(4^\text{th}\) number?
A child collected \(5\) birch leaves, \(2\) oak leaves, and \(5\) maple leaves. What is the mean number of leaves collected for each tree?
There are \(3\) observations, so to calculate the mean we add them up and divide by \(3:\)
\[ \frac{5+2+5}{3} = \frac{12}{3} = 4. \]
Thus the mean number of leaves collected is \(4.\) \( _\square \)
A school collected \(18\) exam papers for the graduating class. \(5\) students received a score of \(10,\) \(4\) students received a score of \(8,\) and \(9\) students received a score of \(7.\) What is the mean score?
There are \( 5 + 4 + 9 = 18 \) total students, so there are \(18\) scores we must average.
Adding up all \(18\) scores, we see that the total of all scores is \( 5 \times 10 + 4 \times 8 + 9 \times 7 = 50 + 32 + 63 = 145 \). The mean, therefore, is
\[ \frac{145}{18} \approx 8.06.\ _\square \]
In Mrs. Mandy's math class, there are \(10\) girls and \(20\) boys. In a recent test, the mean of the girls was \(80,\) while the mean of the boys was \(86.\) What is the mean of the class?
We might be tempted to say that the mean is \( \frac{ 80 + 86 } { 2} = 83 \), because there are those 2 scores. However, this does not work, because there are different numbers of students in each group. Instead, we have to find the total number of students and their total score.
There are a total of \( 10 + 20 = 30 \) students. The girls' total score is \( 10 \times 80 = 800 \) and the boys' total score is \( 20 \times 86 = 1720\). Hence, the total score is \( 800 + 1720 = 2520 \).
As such, the mean of the class is \( \frac{ 2520 } { 30 } = 84 \). \(_\square\)
Weighted Average
A weighted average is an average that takes into account the importance of certain values in a set of data. This is done by taking the product of each element in the data set with its respective "weight," summing these products, and then dividing by the sum of weights.
On a test, \(5\) students received a grade of \(100\), \(3\) students received a grade of \(90\), \(6\) students received a grade of \(80\), \(4\) students received a grade of \(70\), and \(2\) student received a grade of \(60\).
What is the weighted average of the grades?
The possible grades are \(100,\) \(90,\) \(80,\) \(70,\) and \(60,\) but not all grades have the same "weight" since the distribution of grades isn't uniform. To find the weighted average, we multiply each grade by the number of students who received that grade, we sum these products, and then we divide by the total number of students:
\[\frac{(100)(5)+(90)(3)+(80)(6)+(70)(4)+(60)(2)}{5+3+6+4+2} = 82.5.\ _\square\]
Different Types of Series
The definition of arithmetic mean varies slightly depending on the type of series we are averaging:
Individual Series: Take the sum of all observations and divide by the number of observations to get the mean.
Discrete Series: \(\sum {xf}/\sum {f},\) where \(x\) is the value of the observation and \(f\) is the frequency of the observation.
Continuous Series: \(\sum {xf}/\sum {f},\) where \(x\) is the class mark or the midpoint of the class interval and \(f\) is the corresponding frequency of the class interval.
For more information on statistical series, see Statistical Series.