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 the mean is written and pronounced "x bar." To calculate the mean, we take the sum of all values and divide by the number of values. Specifically,
Note that if the mean is calculated by sampling from a larger population, it is called the sample mean, to distinguish it from the population mean denoted or 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.
What is the mean of the values
There are a total of values.
The mean is .
Use the concept of mean to solve the following problem:
A child collected birch leaves, oak leaves, and maple leaves. What is the mean number of leaves collected for each tree?
There are observations, so to calculate the mean we add them up and divide by
Thus the mean number of leaves collected is
A school collected exam papers for the graduating class. students received a score of students received a score of and students received a score of What is the mean score?
There are total students, so there are scores we must average.
Adding up all scores, we see that the total of all scores is . The mean, therefore, is
In Mrs. Mandy's math class, there are girls and boys. In a recent test, the mean of the girls was while the mean of the boys was What is the mean of the class?
We might be tempted to say that the mean is , 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 students. The girls' total score is and the boys' total score is . Hence, the total score is .
As such, the mean of the class is .
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, students received a grade of , students received a grade of , students received a grade of , students received a grade of , and student received a grade of .
What is the weighted average of the grades?
The possible grades are and 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:
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: where is the value of the observation and is the frequency of the observation.
Continuous Series: where is the class mark or the midpoint of the class interval and is the corresponding frequency of the class interval.
For more information on statistical series, see Statistical Series.