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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 4

Measures of Dispersion

Learning Objectives

 Explain the purpose of measures of dispersion, and the information they convey.

 Compute and explain the range (R), the inter-quartile range (Q), the standard deviation

(s), and the variance (s 2

).

 Select an appropriate measure of dispersion and correctly calculate and interpret the

statistic.

 Describe and explain the mathematical characteristics of the standard deviation.

 Analyze a box-plot.

 Use SPSS to produce the standard deviation and range.

Unit Outline

 Using Statistics

 Introduction

 The Range and Interquartile Range

 The Standard Deviation and Variance

 Interpreting the Standard Deviation

 Visualizing Dispersion: Box-plots

Using Statistics

 Measures of dispersion are used to describe the variability or diversity in a set of scores.

They can be used to describe the:

 Variability in hospital patients’ body weight

 Diversity in life styles across different social settings

 Differences in the level of need for health care subsidies across social class levels

 Variations in income inequality across nations over time

Introduction

 Examine these two distributions of ambulance response times. Service A has low

dispersion because it has more similar values. Service B has high dispersion because it

has more dissimilar values.

The Range

 The range (R) is the distance between the highest and lowest scores in a

distribution.

 The range is easy to calculate and serves as a quick measure of variability in a

distribution.

 The greater the value, the more dispersion in the distribution.

Computing the Range

 Arrange the scores from low to high.

 The range is equal to the difference between the lowest value and the highest value.

 Limitation of the range:

 The range is based on only two scores in the distribution, the two most

extreme (the highest, and the lowest).

 The range provides no information on the variation of the scores

between the highest and the lowest values.

The Interquartile Range

 The interquartile range (Q) is the distance between the third quartile (Q 3 ) and first

quartile (Q 1 ) in a distribution.

 The interquartile range provides more information than the range.

 The interquartile range focuses only on the middle 50% of values in a distribution.

Computing the Interquartile

 Arrange the scores from low to high.

valuelowestvaluehighestR 

13 QQQ 

 Find the third quartile (Q 3 ), or the point at which 75% (three-quarters) of a

distribution falls at or below.

 Multiply N by 0.75. This yields the case number of the third quartile.

(If you get a fractional value, round to the nearest case number.)

 Find the first quartile (Q 1 ), or the point at which 25% (one-quarter) of a distribution

falls at or below.

 Multiply N by 0.25. This yields the case number of the first quartile.

(If you get a fractional value, round to the nearest case number.)

 Limitations of the interquartile range:

 Like the range, the interquartile range is based on only two scores.

 However, the interquartile range focuses on the middle, or most

typical, 50% of the values

 The interquartile range ignores the bottom 25% and top 25% of the

distribution.

Computing the Range and Interquartile Range

 Example: Calculate the range and interquartile range for the following

distribution.

The Standard Deviation and Variance

 The ideal measure of dispersion

 Uses all the scores in a distribution.

 Describes the average or typical deviation of the scores.

 Increases in value as the scores become more diverse.

A measure of dispersion focused on deviations (the differences between each individual score

and the mean) meets these three criteria. In the formula below, the deviation from the mean is

calculated for one value.

-The standard deviation (s) describes the dispersion of a distribution.

-The variance (s 2

) is primarily used in inferential statistics



 Note: For both formulas, use N in the denominator for populations and use N-1 in

the denominator for samples.

 To compute the standard deviation (and variance) set up a table with three columns: a

score (x) column, a deviation column, and a deviation-squared column.

Sum up the values in the deviation-squared column – this gives you the numerator. N (for

populations) or N-1 (for samples) is your denominator. Divide these values and then take the

square-root.

 Example: Calculating the dispersion in students’ ages

 xxDeviation i 

 

N

xx s

i  

2

 

N

xx s

i  

2

2

Interpreting the Standard Deviation

 The standard deviation can be interpreted as:

 An index of variability that increases in value as the distribution

becomes more variable.

 The minimum possible value is zero – where there is no

variation or when all cases are the same.

 A method of comparing one distribution with another.

 A tool for determining the area under the normal curve.

Interpreting Box Plots

 Boxplots provide a helpful way to visualize and analyze dispersion.

 Boxplots display the median, the range, and the Interquartile range.

 In this example, each box, and its accompanying lines and “whiskers” (the t-shaped

endings) shows the birth rates per 1,000 population in nations with low, lower middle,

upper middle, and high income.

Summary of Measures of Dispersion

 Measures of dispersion summarize information about the heterogeneity, or variety,

in a distribution of scores. While measures of central tendency locate the central

points of the distribution, measures of dispersion indicate the amount of diversity in

the distribution.

 The range (R) is the distance from the highest to the lowest score in the distribution.

The interquartile range (Q) is the distance from the third to the first quartile (the

“range” of the middle 50% of the scores). These two ranges can be used with

variables measured at either the ordinal or interval-ratio level.

 The standard deviation (s) is the most important measure of dispersion because of

its key role in many more advanced statistical applications. The standard deviation

has a minimum value of zero (indicating no variation in the distribution) and

increases in value as the variability of the distribution increases. It is used most

appropriately with variables measured at the interval-ratio level.

 Boxplots provide a visual way of analyzing dispersion by graphically illustrating

the range, the interquartile range, outliers, and extreme outliers.

Basic Terms

 Boxplot

 Deviation

 Dispersion

 Interquartile range (Q)

 Measures of dispersion

 Range (R)

 Standard deviation

 Variance