Measures of Central Tendency
A measure if central tendency is a single value that attempts to describe a set of data
by identifying the central position within the set of data. As such, measures of central
tendency are sometimes called the measures of central location.
A. Mean
The mean of a set of data is the sum of all values in data set divided by the number
of values in the set. It is often referred to as an arithmetic average. The Greek letter
µ is used as the symbol for population mean and the symbol x̄ is used to represent
the mean of a sample. To determine the mean of a data set.
1. Add together all the data values.
2. Divided the sum obtained from step 1 by the number of data values in the set.
Example:
Consider the data set: 10, 9, 15, 13, 20, 7, 5, 13, 14, 18, 19; find the mean.
Answer:
B. Median
The median of a data set of data is the “middle element” when the data is
arranged in ascending order. To determine the median:
1. Put the data in order from the smallest to largest.
2. Determine the number in the exact center.
i.
ii.
If there are an odd number of data points, the median will be the number
in the absolute middle.
If there is an even number of data points, the median is the mean of the
two center data points, meaning the two center values should be added
together and divided by 2.
Example:
a. Consider the data set:
113, 115, 127, 123, 129, 103, 105, 106, 108, 115, 114, 119, 120
b. Consider the data set:
85, 90, 97, 65, 74, 92, 84, 83, 72, 70, 94, 87, 81, 67
C. Mode
The mode is the most frequently occurring measurement in a data set. There may
be one mode; multiple modes, if more than one number occurs most frequently, or
no mode at all, if every number occurs only once. To determine the mode:
1. Put the data in order from smallest to largest, as you did to find your median.
2. Look for any value that occurs more than once.
3. Determine which of the values from step 2 occurs most frequently.
Example:
Consider data set:
150, 160, 155, 153, 165, 170, 154, 150, 154, 143, 142, 162, 172, 143, 132, 165, 123,
144, 155, 132, 122, 167, 152, 142, 135, 137, 173, 150, 122, 146, 178, 175, 161, 159,
143, 121, 113, 127, 129, 151
Measures of Dispersion
A measure of dispersion indicates the scattering of data. It explains the disparity of the
data from one another, delivering precise view of their distribution from one another.
The measure of dispersion displays and gives us an idea about the variation and the
central value of an individual item
A. Range
The range is the difference between the lowest and highest value in a data set.
To determine the range:
1. Identify the largest value in your data set. This is called the maximum
2. Identify the lowest value in your data set. This is called the minimum
Example:
Consider the data set:
145, 134, 178, 198, 123, 156, 187, 154, 132, 137, 192, 125, 183, 164, 178, 155, 166,
143, 137, 132, 176, 144, 177, 152.
B. Variance and Standard Deviation
The variance and standard deviation are a measure based on the distance each
value is from the mean.
1.
2.
3.
4.
5.
6.
7.
Find the mean of the data.
Subtract the mean from each of the data value.
Square each calculation from step 2.
Add the values of the squares from step 3.
Find the number of data points in your set, called n.
Divide the sum from step 4 by the number n (if population) or n-1 (if sample)
To find the standard deviation, square root the calculated result obtained in
step 6.
C. Upper and Lower Quartiles
The quartiles of a group of data are the medians of the upper and lower halves
of that set. The lower quartile, Q1, is the median of the lower half, while the
upper quartile, Q3, is the median of the upper half. If the data set has an odd
number of data points, you do not consider your median when finding these
values, but if the data set contains an even number of data points, you will
consider both middle values that you used to find the median as parts of the
upper and lower halves.
To find the upper and lower quartile:
1.
2.
3.
4.
Put the data in order from smallest to largest.
Identify the upper and lower halves of the data.
Using lower half, find Q1 by finding the median of that half.
Using upper half, find Q3 by finding the median of that half.