In statistics, the mean of a set has many definitions, the arithmetic mean, the geometric mean, the harmonic mean, and the population mean.

The arithmetic mean is what we all are already familiar with known as the average of a given set and is generally plainly called the mean. In order to find the arithmetic mean, one must take the sum of the numbers of a set and divide by the amount of variables in that set.

SAT question: Arithmetic Mean
The average (arithmetic mean) of 3 numbers is 60. If two of the numbers are 50 and 60, what is the third number?
a). 50
b). 55
c). 60
d). 65
e). 70

Answer is E. We know that the sum of the 3 numbers will be 180 and we also know that two of the numbers are 50 and 60, so we subtract them from 180.

The geometric mean is the set of positive numbers that is interpreted in accordance to their product not their sum. In order to calculate the geometric mean, one must multiply all of the numbers of a set and take the product and have it to the power of one over how many variables in the set.

SAT question: Geometric Mean
What is approximately the geometric mean of the given set [5, 7, 2, 1]
a). 5.630
b). 2.5
c). 3.152
d). 2.893
e). 1.16

The answer is D. We multiply the numbers in the given set, which is 70, and take it to the 1/4 power because there was 4 variables in that set.

The harmonic mean is an average for a set of numbers for a specific unit such as speed or energy. To find the Harmonic mean of a given set of values, one must take the number of variables and divide it by the reciprocals of the sum of those variables


Harmonic Mean: SAT Question
The distance from Tim's house to town is 40 km. He drove to town at a speed of 40 km per hour and returned home at a speed of 80 km per hour. What was Tim's approximate average speed for the whole trip?
a). 53.33
b). 49.31
c). 3.14
d). 60
e). 56.7
The answer is A. We know that the two speeds are 40 km/hour and 80 km/hour, and we need to find the harmonic mean. So it would be 2 divided by the sum of the reciprocals of the two values. So it will be approximately 53.33


The Median of a set of data is a number that separates the data so that at most 50% of the data is above the median, and at most 50% is below the median.
To find the median of a set of data, one must first order the data in numerical order. If there are an odd amount of data points, the median is the number exactly in the middle. For and even amount of data points, the median is the average of the two middle numbers.
To find the middle number for a large set of data, add 1 to the total data points, and divide it by two. This number is the xth number in order, and is the median.

Median would be the most useful in certain situations. In a case where there are a few outliers, this would skew the average. For example, if the ages of people in an apartment are 60, 75, 68, 64 and 21, the 21 would skew the average to a younger age (57.8). The median would be 64, which is higher than the average, and a better representation of the "middle value" of the data.

Medians are readily apparent in Box Plots where the line dividing the box into two is the median:

Simple Example and Explanation:
Find the median of the following test scores:

72, 93, 86, 70, 82, 68, 86, 92, 98, 87, 86

Answer: 86
Explanation: First order the data from smallest to largest:
65, 71, 73, 81, 84, 86, 89, 90, 92, 95, 98
There are 11 numbers, therefore the middle number is the sixth number [(11+1)/(2)=6], which is 86


Mode – Is the value that occurs the most in a set of data or probability distribution.
There can be more than mode.

Example 1:
Mr. Killian’s Honors Math Analysis test grades are:
72, 93, 86, 70, 82, 68, 86, 92, 98, 87, 86

Order the data from smallest to largest
68, 70, 72, 82, 86, 86, 86, 87, 92, 93, 98

The test grade that occurs the most is 86 so the mode is 86.

The mode of a discrete probability distribution is the value that is most likely to result.
Example 2:
The mode of a histogram is a relative maximum.
In this example the values -6 and 6 both result in a frequency of 3, therefore the modes of this histogram are: -6 and 6.

Example 3
A coin is tossed two times. Let x be the number of tails which can come up. The sample space is given below. What is the mode of this probably distribution?


The number of tails most likely to appear is 1 so therefore the mode is 1.

Example 4
If a set of numbers contains the following values:
{12, 37, 38, 42, 78, 81}
What would be the mode?

There would be no mode in this situation, because the frequency of each number is one. No number appears more often than another.

The mode of a continuous probability distribution graph is the x value that achieves the maximum y value, in other words the peak or highest point in the graph.

Example 5
What is the mode of the graph above?
When x is 8 the maximum y value is achieved therefore the mode of the graph is 8.

Mega Question

The table above represents the number of admission tickets that were sold by an amusement park over a seven day period. If x represents the average number of tickets sold, y represents the median number of tickets sold, and z is the number of tickets that were sold most often, which of the following correctly expresses the relationship between these values?
A. x > y > z
B. z > x > y
C. y > x > z
D. x > z > y
E. z > y > x

A. x > y > z

Mean (x): (622+801+1368+801+1675+1914+2565)/7 = 1392.286...
Median (y): (622, 801, 801, 1368, 1675, 1914, 2565; middle number is (7+1)/2 = 4th number) 1368
Mode (z): 622 (Appears twice, no other value is repeated)
1392 > 1368 > 622
x > y > z
Letter A is the answer

Websites (for box plot example)