measures of central tendency & dispersion research methods

MEASURES OF CENTRAL TENDENCY& DISPERSION

Research MethodsResearch Methods

Vocabulary Review

Adopted from Monica Yuskaitis, 2000

Sum – the answer to an addition problem.

Addend – the numbers you added together to get the sum.

6 + 9 = 15

Definition

MeanMean

MeansMeans

AverageAverage

Definition

MeanMean – the average of a group of numbers.

2, 5, 2, 1, 5Mean = 3Mean = 3

Formula for the Mean

Copyright © 2000 by Monica Yuskaitis

The Mean is found by evening out the numbers

2, 5, 2, 1, 5

Mean is found by evening out the numbers

2, 5, 2, 1, 5

Mean is found by evening out the numbers

2, 5, 2, 1, 5mean = 3

How to Find the Mean of a Group of Numbers

Step 1 – Add all the numbers.

8, 10, 12, 18, 22, 26

8+10+12+18+22+26 = 96


Step 2 – Divide the sum by the number of addends.

8, 10, 12, 18, 22, 26

8+10+12+18+22+26 = 96 How many addends are there?


Step 2 – Divide the sum by the number of addends.

6)96 sum# of addends1

636

6

63


The mean or average of these numbers is 16.

8, 10, 12, 18, 22, 26

What is the mean of these numbers?

7, 10, 16

11


2, 9, 14, 27

13


1, 2, 7, 11, 19

8


26, 33, 41, 52

38

Definition

MeMeddianian

is in theis in the

MiMiddddlele

Definition

MedianMedian – the middle number in a set of ordered numbers.

1, 3, 7, 10, 13Median = 7Median = 7

How to Find the Median in a Group of Numbers

Step 1 – Arrange the numbers in order from least to greatest.

21, 18, 24, 19, 27

18, 19, 21, 24, 27


Step 2 – Find the middle number.

21, 18, 24, 19, 27

18, 19, 21, 24, 27


Step 2 – Find the middle number.

18, 19, 21, 24, 27

This is your median number.


Step 3 – If there are two middle numbers, find the mean of these two numbers.

18, 19, 21, 25, 27, 28


Step 3 – If there are two middle numbers, find the mean of these two numbers.

21+ 25 = 46

2) 46 23 median

What is the median of these numbers?

16, 10, 7

10

7, 10, 16


29, 8, 4, 11, 19

114, 8, 11, 19, 29


31, 7, 2, 12, 14, 19

132, 7, 12, 14, 19, 31

12 + 14 = 26 2) 26


53, 5, 81, 67, 25, 78

6053 + 67 = 120 2) 120

5, 25, 53, 67, 78, 81

Definition

ModeMode

is the mostis the most

PopularPopular

Definition

A la modeA la mode – the most popular or that which is in fashion.

Baseball caps are a la mode today.

Definition

ModeMode – the number that appears most frequently in a set of numbers.

1, 1, 3, 7, 10, 13Mode = 1Mode = 1

How to Find the Mode in a Group of Numbers


21, 18, 24, 19, 18

18, 18, 19, 21, 24

How to Find the Mode in a Group of Numbers

Step 2 – Find the number that is repeated the most.

21, 18, 24, 19, 18

18, 18, 19, 21, 24

Which number is the mode?


29, 8, 4, 8, 19

84, 8, 8, 19, 29



1, 2, 2, 9, 9, 4, 9, 10

9

1, 2, 2, 4, 9, 9, 9, 10


22, 21, 27, 31, 21, 32

21

21, 21, 22, 27, 31, 32

Definition

RangeRange

is the distanceis the distance

BetweenBetween

Definition

RangeRange – the difference between the greatest and the least value in a set of numbers.

1, 1, 3, 7, 10, 13Range = 12Range = 12

How to Find the Range in a Group of Numbers


21, 18, 24, 19, 27

18, 19, 21, 24, 27


Step 2 – Find the lowest and highest numbers.

21, 18, 24, 19, 27

18, 19, 21, 24, 27


Step 3 – Find the difference between these 2 numbers.

18, 19, 21, 24, 27

27 – 18 = 9The range is 9

What is the range?

29, 8, 4, 8, 19

29 – 4 = 25

4, 8, 8, 19, 29

What is the range?

22, 21, 27, 31, 21, 32

32 – 21 = 11

21, 21, 22, 27, 31, 32

What is the range?

31, 8, 3, 11, 19

31 – 3 = 28

3, 8, 11, 19, 31

What is the range?

23, 7, 9, 41, 19

41 – 7 = 34

7, 9, 23, 19, 41

Variance

the variance of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.

Calculating Variance

The variance is computed as the average squared deviation of each number from its mean. For example, for the numbers 1, 2, and 3, the mean is 2 and the variance is

Variance in a Sample

The Variance of a sample is given by replacing the expected value with the mean of the sample

Unbiased Estimate of Sample Variance

However, the previous equation is biased, since it does not allow for the fact that a parameter (the mean) is fixed.

Hence, the unbiased estimate of the sample variance employs N-1 in the denominator.

Since samples are usually used to estimate parameters, s² is the most commonly used measure of variance. Calculating the variance is an important part of many statistical applications and analyses.

Standard Deviation

The Standard Deviation divides a distribution into standard units represented as deviations from the mean.

One standard deviation away from the mean in either direction on the horizontal axis (the red area on the above graph) accounts for somewhere around 68 percent of the people in this group. Two standard deviations away from the mean (the red and green areas) account for roughly 95 percent of the people. And three standard deviations (the red, green and blue areas) account for about 99 percent of the people.

Standard Deviations

Suppose your data follows the classic bell shaped curve pattern. One conceptual way to think about the standard deviation is that it is a measures of how spread out the bell is.

Shown here is a bell shaped curve with a standard deviation of 1. Notice how tightly concentrated the distribution is.

Standard Deviations

Shown here is a different bell shaped curve, one with a standard deviation of 2. Notice that the curve is wider, which implies that the data are less concentrated and more spread out.

Standard Deviations

Finally, a bell shaped curve with a standard deviation of 3 appears below. This curve shows the most spread.

Equation for the Standard Deviation of a Sample

Let’s calculate the standard deviation of a set of data.

Calculating Standard Deviations

(73-69.9)2 = (3.1)2 = 9.61

(58-69.9)2 = (-11.9)2 = 141.61

(67-69.9)2 = (-2.9)2 = 8.41

(93-69.9)2 = (23.1)2 = 533.61

(33-69.9)2 = (-36.9)2 = 1361.61

(18-69.9)2 = (-51.9)2 = 2693.61

(147-69.9)2 = (77.1)2 = 5944.41

The Seven values in this data set are 73, 58, 67, 93, 33, 18, and 147. The mean for this data set is 69.9.

For each data value, compute the squared deviation by subtracting the mean and then squaring the result.

The sum of these squared deviations is 10,692.87. Divide by 6 to get 1782.15. Take the square root of this value to get the standard deviation, 42.2.

The seven

Standard Deviation


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