Chapter 2 Review

• Using graphs/tables/diagrams to show variable relationships

• Understand cumulative frequency, percentile rank, and cross-tabulations

• Perform rates of change

Cumulative Frequencies / Percentile Rank

# of Arrests f % cf C%

9-11 1 3.3 30 100

6-8 5 16.6 29 96.5

3-5 8 26.6 24 79.9

0-2 16 53.3 16 53.3

What is the percentile rank for those with 5 arrests?

Cross-Tabulations Attitude towards Lowering the Drinking Age to 19

Male Female Total

Favor 584 765 1349

27% 23% 24.5%

Neutral 498 902 1400

23% 27% 25.5%

Oppose 1082 1667 2749

50% 50% 50%

Total 2164 3334 5498

100% 100% 100%

Rate of Change• Rate of Change =

(100) * (time 2f – time 1f) (time1f)

• Allows us to compare the same population at two points in time.

• Always be aware of the sign. – A negative percent signifies a reduction– A positive percent signifies an increase

Chapter 3Measures of Central Tendency

Measures of Central Tendency

• Three main types– Mode– Median– Mean

• Choice depends upon level of measurement

The Mode• The mode is the most frequently occurring value

in a distribution.• Abbreviated as Mo• Sometimes there is more than one mode • EX: 96, 91, 96, 90, 93, 90, 96, 90• Bimodal• Mode is the only measure of central tendency

appropriate for nominal-level variables

Mode - Example• What is the mode for the following set of

numbers?• 20, 21, 30, 20, 22, 20

• Explains nothing about– Ordering of variables– Variation within variables

• Distributions can be bimodal and/or multimodal– Several categories with same frequencies

The Median• The median is the middle case of a distribution• Abbreviated as Mdn• Appropriate for ordinal data because it only shows

direction and not distance• Used if distribution is skewed• How to find the median?

• If even, there will be two middle cases – interpolate• If odd, choose the middle-most case

• Cases must be ordered

21

N

Position of the Mdn

Example of median: Years in Prison• What is the median?

– odd or even?

• (7+1)/2=4th case• Where is the 4th case?• Sort distribution from lowest to highest

• 1• 5• 2• 9• 13• 11• 4

Example of median with even # of cases

• (8+1)/2=4.5 • Half way between the 4th and

5th case• (2 + 3) / 2 = 2.5• Median = 2.5

• 1• 1• 2• 2• 3• 4• 4• 6

21

N

Position of the Mdn

The Mean• Most popular measure of central tendency• Assumes equality of intervals• Basis of many higher order formulas for

statistical procedures• Use either μ or X depending on whether

population or sample estimate

The Mean• The mean is

appropriate for interval and ratio level variables

NX

X meanX sum

X = raw scores in a set of scoresN = total number of scores in a set

Example: Prison Sentences• What is the mean?• 4.6

• 7.9• 11.4• 2.2

NX

X

53.64

)2.24.119.76.4(

X

yearsX 5.6

The Mean

• What does the mean do?– Center of gravity– Deviation =

(Raw Score – Mean)

X (Raw Score) Deviation

9 +3

8 +2

6 0

5 -1

2 -4

Mean = 6 = (∑X / N)= (30 / 5)

The Weighted Mean• The “mean of the means” – overall mean for a number of groups

• Best used for unequal groups

Example:4, 7, 3, 82, 4, 9, 1, 6, 8

An Illustration: Measures of Central Tendency in a Skewed Distribution

Salary$120,000$60,000$40,000$40,000$30,000$30,000$30,000

Mean = $50,000

Median = $40,000

Mode = $30,000

Comparing the Mode, Median, and Mean• Three factors in choosing a measure of central

tendency1. Level of measurement2. Shape or form of the distribution of data• Skewness• Kurtosis

3. Research Objective

Level of MeasurementLevel of measurement

Mode

Median

Mean

Nominal Yes

Ordinal Yes Yes

Interval Yes Yes Yes

Shape of the Distribution• In symmetrical distribution – mode, median,

and mean have identical values• In skewed data, the measures of central

tendency are different– Skewness relevant only at the interval level

• Mean heavily influenced by extreme outliers – median best measure in this situation

Research Objective• Choice of reported central tendency depends on the

level of precision required.• Most published research requires median and/or

mean calculations.• In skewed data, median more balanced view• For advanced statistical analyses, mean usually

preferred• In large data sets, mean most stable measure

Summary

• Three best known measure of central tendency – mode, median, mode

• Three factors determine appropriateness– Level of measurement– Shape of the distribution– Research objective