# Basic Statistics

A statistic is a a quantity calculated from a set of data. Useful statistics help describe the characteristics of a data set. For the COMPASS test you’ll want to be familiar with three basic statistics: the mean, median, and mode.

Mean

A mean is the average of a set of numbers. The mean is calculated by summing all of the numbers then dividing this total by the amount of numbers in the data set. To demonstrate, we’ll calculate the mean of the following data set. This data set contains the test scores of a group of 15 test takers.

$\text{Scores:\;} 55,68,71,72,72,72,76,80,84,84,88,90,90,95,98$

Adding up all of these scores gives us 1,195. If we divide 1,195 by 15 we get our average score of 79.7.

Try calculating the mean of this data set: $16,19,19,20,21,25$. You should get 20.

Median

If you sort a data set in ascending or descending order, the number in the middle is the median. The data set of test scores from above is already sorted. The middle number in the list is 80, which is the median.

The second data set is also sorted, but it has an even number of data points so their is no middle number in the list. When this occurs, the median is calculated by averaging the two ‘middle numbers,’ in this case: 19 and 20. Thus our median is: $(19+20)/2=19.5$.

Data sets will not always be written in order as the two above were. Try computing the median of the following data set:

$\text{Points:\;} 8,5,12,23,8,12$

You should get a median of $(8+12)/2=10$.

Mode

The mode is the most frequently occurring point in a data set. In the ‘Scores’ data set our mode is 72, because 72 occurs 3 times in the data set–more than any other number. It is possible for a data set to have two modes (bi-modal). The ‘Points’ data set above has two modes: 8 and 12. When a data set has more than two modes, we usually say there is no mode. The following data set gives an example.

$\text{Ages:\;} 2,7,16,11,8,19$

Each data point only occurs once. There is ‘no mode.’

Practice Questions

For the following data sets calculate the mean, median, and modes:

1.) $\text{Weights:\;} 118,190,145,230,130,145,155$

2.) $\text{Rankings:\;} 9,88,23,43,52,31$

3.) $\text{Speeds:\;} 5.00,4.82,4.47,4.78,4.40,5.20,4.82,4.51$

Solve:

4.) Mrs. Ryan knows the average score of her class on a recent test was 80, but she has misplaced one student’s test. The scores for the other students are: 69,74,76, and 85. What score must be on the missing test?

5.) The average of a data set is 18.3. There are 30 points in the set. If the points 19.8 and 20 are added to the data set, what will the new mean be?