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Unit 3

Measures of Central Tendency

Learning Objectives

 Explain the purposes of measures of central tendency and interpret the information they

convey.

 Calculate, explain, and compare and contrast the mode, median, and mean.

 Explain the mathematical characteristics of the mean.

 Select an appropriate measure of central tendency according to level of measurement and

skew.

 Use SPSS to produce means, medians, and modes.

Unit Outline

 Using Statistics

 Introduction

 The Mode

 The Median

 The Mean

 Three Characteristics of the Mean

 Choosing a Measure of Central Tendency

Using Statistics

 Measures of central tendency are used to find the typical case or average score on a single

variable. They can, for example:

 Identify the most commonly purchased car in the United States.

 Compare public opinion on the Affordable Health Care act over time.

 Measure the median income in Detroit, Michigan.

 Track changes in age at first birth over time.

Introduction

 There are three measures of central tendency. Each one is a way of describing a typical

case or average score in a distribution. They include:

 The mode

 The median

 The mean

 The mode of a distribution is the value that occurs most frequently.

 The mode is most useful when working with nominal level variables.

 The mode is the only measure of central tendency appropriate for nominal

variables.

 Limitations of the mode:

 Some distributions have no mode at all. This occurs when no value occurs more

than once, or when all values occur at the same frequency.

 Some distributions have multiple modes. This occurs when more than one value

(but not all of the values) occur at the same frequency.

 The mode for an ordinal or interval-ratio variable may not be central to the

distribution as a whole.

 The mode can be found in these two distributions by identifying the value with the

highest frequency. In Example A, the mode is Protestant. In Example B. the mode is 93.

 The median (Md) is always at the exact center of a distribution.

 Half of the scores in a distribution are higher than the median, and half of the scores are

lower than the median.

 Calculating the median

 The median can be calculated for ordinal and interval-ratio level data.

 Before determining the median, all of the scores must be arranged in order, from low to

high.

 When there are an odd number of cases (N):

 The median is the exact middle case. Find the case number of the median

by using the formula below.

Example:

The Median is 7.

 When there are an even number of cases (N):

 The median is the value between the two middle-most

cases. Find the case number of the median by adding the

two middle cases and dividing by two.

 Find the first middle case by dividing N by 2. The first

middle case below is 7. The second middle case is the

next case. In the example below, the second middle case

is 5. Add 7 plus 5 and divide by two.

 Example:

2

1N

The median is 6.

 Limitations of the median:  Nominal variables do not have a median because their categories

cannot be ranked from low to high.

 Although the median is the centermost score, it represents only one

point in the data. It may not be very representative of the other scores.

 The mean is the arithmetic average of all scores in a distribution.

 The mean is the most commonly used measure of central tendency, but

it can only be used for interval-ratio level data.

-A sample mean is denoted with the symbol:

-A population mean is denoted with the symbol:

 Calculating the mean:

Where:

x is the sample mean

is the Summation of all the scores

x 

N

x x

i 

 ix

N is the total number of scores

Example

Calculate the mean of these grades on homework assignments:

85, 92, 78, 86, 94, 80

85 + 92 + 78 + 86 + 94 + 80 = 515 = 85.83 = 14.31

6 6

 Limitations of the mean:  The mean is most appropriate for interval-ratio variables.

 However, because of some useful properties of the mean, it is

sometimes calculated on ordinal variables.

 The mean can be deceiving when data are skewed

Characteristics of the Mean

-The mean balances all the scores. -The mean is like a fulcrum that “balances” all scores in a distribution.

-The mean is the central point around which all scores “cancel out” each other. One low score,

cancels out a high score

 The mean minimizes the variation of the scores.  This is also called the “least squares” principle of the mean.

 The mean is the point in a distribution around which the variation (or

differences) in scores is minimized.

 The mean is closer to all of the scores in a distribution than any other

measure of central tendency.

 The mean can be misleading if the distribution is skewed.

N

x x

i 

   0xxi

   minimum 2

xxi

 A “skew” occurs when an otherwise normal distribution has a few extremely high

or a few extremely low scores.

 Positively skewed distributions have a few high scores that “pull” the

mean higher.

 Negatively skewed distributions have a few low scores that “pull” the

mean lower.

 Skewed data can best be better described with the median.

 Below are two examples of skewed distributions.

 In a normal distribution (unskewed, symmetrical), the mode, median and mean

will all be equal.

Choosing a Measure of Central Tendency

 Two main criteria are used when choosing a measure of central tendency:

 The level of measurement

 For nominal variables, only the mode is possible.

 For ordinal variables, the mode and median are possible,

but the median is usually preferred (except in some

inferential statistics, when the mean is used).

 For interval-ratio variables, all three measures are

possible, but the mean is usually preferred (except in

cases of skewness, when the median is used).

Unit 5: The Normal Curve

Learning Objectives

 Define and explain the concept of the normal curve.

 Convert empirical scores to Z scores and use Z scores and the normal curve table

(Appendix A) to find areas above, below, and between points on the curve.

 Express areas under the curve in terms of probabilities.

Unit Outline

 Using Statistics

 Properties of the Normal Curve

 Using the Normal Curve

 Using the Normal Curve to Estimate Probabilities

Using Statistics

 Using Statistics

 Properties of the Normal Curve

 Using the Normal Curve

 Using the Normal Curve to Estimate Probabilities

Properties of the normal Curve

 The normal curve is a theoretical model.

 The normal curve is unimodal, perfectly smooth, and symmetrical; the mode, median,

and mean are the exact same value.

 The normal curve is bell-shaped and has “tails” that extend infinitely in both directions.

 Distances along the horizontal axis are measured in standard deviation units.

 When measured in standard deviations from the mean, distances along the horizontal axis

will always encompass the same proportion of the total area under the curve.

 The proportion of cases falling between the mean and one standard deviation

above the mean on one variable will be the exact same as the proportion of cases

falling between the mean and one standard deviation above the mean on a

different variable.

 68.26% of the area under the normal curve is found between 1 standard deviation below

the mean, and 1 standard deviation above the mean.

 95.44% of the area under the normal curve is found between 2 standard deviations below

the mean, and 2 standard deviation above the mean.

 99.72% of the area under the normal curve is found between 3 standard deviations below

the mean, and 3 standard deviation above the mean.

 Areas under the normal curve can be expressed as the proportion of area, the

number of cases, or the probability of a case, falling above some point, below some

point, or between two points.

Using the Normal Curve

 To use the normal curve, we calculate Z scores.

 Z scores are “raw” values that have been “standardized” (or converted) into

standard deviation units.

 A distribution of Z scores will have a mean of zero (0) and a standard deviation of

one (1).

Computing Z- Scores

 Z scores are measured in units of standard deviation.

 A score that falls one standard deviation above the mean will have a Z

score of +1.

 A score that falls two standard deviations below the mean will have a

Z score of -2.

 Z scores can take on fractional values; for example, a score can fall

2.75 standard deviations above the mean.

s

xx Z i 

 The normal curve table presents all possible areas under the normal curve for each

Z score.

 For each Z score, the table provides the proportion of the area:

 Between the mean and that particular Z score (the B Column)

 From that Z score and away from the mean (the C Column)

 Because the normal curve is symmetrical, the normal curve table only includes

positive Z scores. The areas for negative Z scores will be identical.

 The total area above a mean is 0.50 (50%); the total area below a mean is 0.50

(50%). The total area under the entire curve is 1.00 (100%).

 The normal curve table presents all possible areas under the normal curve for each Z

score.

 For each Z score, the table provides the proportion of the area:

 Between the mean and that particular Z score (the B Column)

 From that Z score and away from the mean (the C Column)

 Because the normal curve is symmetrical, the normal curve table only includes positive Z

scores. The areas for negative Z scores will be identical.

 The total area above a mean is 0.50 (50%); the total area below a mean is 0.50 (50%).

The total area under the entire curve is 1.00 (100%).

 Finding the Total Area Above or Below a Score

a. Calculate the Z score.

b. Find the Z score in the normal curve table.

c. Use the B and C columns of the table to determine the area.

 Finding the Area Below a Positive Z Score*

 In a distribution with a mean of 100 and a standard deviation of 20,

find the area below 108 by:

a) calculating the Z score (.40) (see below);

b) finding the Z score in the normal curve table; &

c) reporting the area in the B Column + .50

(.1554+.50=.6554).

 Multiply by 100 to convert your answer to a

percentage (65.54%)

*scores above the mean have a Positive Z Score

 Finding the Area Below a Negative Z Score*

 In a distribution with a mean of 100 and a standard deviation of 20,

find the area below 80 by:

a) calculating the Z score (-1.00) (see below);

b) finding the Z score in the normal curve table; &

c) reporting the area in the C Column (.1587)

 Multiply by 100 to convert your answer to a

percentage (15.87%)

*scores below the mean have a Negative Z Score

 Finding Areas Between Two Scores

a. Calculate the Z score for both scores.

b. Find the Z scores in the normal curve table.

Use the B and C columns of the table to determine the area.

 Finding the area between a Negative Z Score and a Positive Z Score.

 In a distribution with a mean of 100 and a standard deviation of 20,

find the area between 93 and 112 by:

a) calculating the Z scores (-.35 and .60) (see below);

b) finding the Z scores in the normal curve table; &

c) reporting the sum of the areas in the B Columns

(.1368+ .2257 = .3625)

Multiply by 100 to convert your answer to a percentage (36.25

40. 20

8

20

100108 

 

 

s

XX Z t

00.1 20

20

20

10080 

 

 

s

XX Z t

 Finding the area between two Negative Z Scores or two Positive Z Scores.

 In a distribution with a mean of 100 and a standard deviation of 20,

find the area between 113 and 121 by:

a) calculating the Z scores (+.65 and +1.05) (see below);

b) finding the Z scores in the normal curve table; &

c) reporting the difference of the areas in the B Columns

(.3531 - .2422 = .1109) or the C Columns (.2578 - .1469 = .1109)

Multiply by 100 to convert your answer to a percentage (11.09%)

Using the Normal Curve to Estimate Probabilities

 The normal curve can also describe the probability (or likelihood) that a score will

fall above, below, or between scores in a distribution.

 The same steps used in the previous four slides can be used to calculate

probabilities that a value will fall above a score, below a score, or between two

scores.

 Note that the probability of randomly selecting a value close to the mean (+/- 1

standard deviation) is greater than that of selecting a value further away from the

mean.

Summary

 The normal curve can also describe the probability (or likelihood) that a score will

fall above, below, or between scores in a distribution.

 The same steps used in the previous four slides can be used to calculate

probabilities that a value will fall above a score, below a score, or between two

scores.

 Note that the probability of randomly selecting a value close to the mean (+/- 1

standard deviation) is greater than that of selecting a value further away from the

mean.