Measures of a Distribution

1
Measures of a Distributions Central Tendency,
Spread, and Shape
SHARON LAWNER WEINBERG SARAH KNAPP ABRAMOWITZ
Statistics SPSS An Integrative
Approach SECOND EDITION
Using

• Chapter 3

2
Overview

• Measures of Central Tendency (Level)
• Mode
• Median
• Mean
• Measures of Dispersion (Spread)
• Range
• Interquartile Range
• Variance
• Standard Deviation
• Measure of Shape
• Skewness and Skewness Ratio

3
Measures of Central Tendency Mode

• Definition The mode is the score that occurs
  most often.
• Useful when data are nominal or ordinal with
  only a limited number of categories.
• To find the mode, click Analyze on the main menu
  bar,
• Descriptive Statistics, and then Frequencies.
• Click on Options, and the square next to mode.
  Click OK.

4
Measures of Central Tendency Mode

• Example Home Language Background (HOMELANG)
• What is this variables mode?

Mode English Only
5
Measures of Central Tendency Mode

• Example Although the mode is technically the
  South, the North Central is close enough that the
  distribution may be considered bimodal.

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Measures of Central Tendency Mode

• Definition A bimodal distribution is one with
  two modes, usually at some distance apart from
  each other.
• Definition A uniform distribution is one in
  which all values occur with the same frequency.

7
Measures of Central Tendency Median

• Definition The median is the middle point in a
  distribution.
• Useful when data are ordinal or scale and
  severely skewed.
• To find the median, click Analyze on the main
  menu bar,
• Descriptive Statistics, and then Explore. Click
  OK.
• Or, to find the median, click Analyze on the main
  menu bar,
• Descriptive Statistics, and then Frequencies.
• Click on Options, and the square next to median.
  Click OK.

8
Measures of Central Tendency Mean

• Definition The mean is the sum of all of the
  data points divided by the number of data points.
• Useful when data are scale and not severely
  skewed.
• To find the mean, click Analyze on the main menu
  bar,
• Descriptive Statistics, and then Explore. Click
  OK.
• OR use Frequencies OR use Descriptives.

9
Measures of Central Tendency Mean

• In the case where the variable is dichotomous and
  coded as 0 and 1, the mean is interpreted as the
  proportion of 1s in the distribution.
• Example Gender

10
Measures of Central TendencyComparing the Mean,
Median, and Mode

• Compare the values of the mode, median, and mean
  for SES, EXPINC30, and SCHATTRT.

11
Measures of Dispersion Visually

• When traveling to these two cities, would the
  same clothing be suitable for both cities at any
  time during the year from the point of view of
  warmth?

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Measures of Dispersion

• How can we quantify the obvious difference in
  temperature variability across the year between
  these two cities?
• One Answer By using the range or interquartile
  range (IQR).
• Another Answer By using the variance or standard
  deviation.

13
The Range and Interquartile Range

• Definition The range is the difference between
  the highest and lowest values in the
  distribution. The interquartile range (IQR) is
  the range of the middle half of the data, or the
  difference between the 75th and 25th percentiles.
• Useful when data are ordinal or scale and
  severely skewed.
• To find the IQR and range, click Analyze on the
  main menu bar, Descriptive Statistics, and then
  Explore. Click OK.

14
The Variance and Standard Deviation

• Definition The variance is the average of the
  squared deviations from the mean. The standard
  deviation is the square root of the variance. We
  may think of the standard deviation as the
  distance we have to travel in both directions
  from the mean to capture the majority of values
  in a distribution. The farther out we need to
  travel, the more spread out are the values of the
  distribution from the mean.
• Useful when data are scale and not severely
  skewed.
• To find the SD and Variance, click Analyze on the
  main menu bar, Descriptive Statistics, and then
  Explore. Click OK.

15
Measures of Dispersion

• We get the following values for the
  temperature example. Consistent with the earlier
  boxplots, for all quantitative measures,
  Springfield is shown to have a greater
  temperature spread than San Francisco.

16
Measures of Dispersion

• Key words to indicate that a question relates to
  dispersion
• Spread, variability, dispersion, heterogeneity,
  inconsistency, unpredictability

17
Measures of Shape

• Definition The skewness statistic is a measure
  of the shape of a distribution. It is negative
  when the distribution is negatively skewed, zero
  when the distribution is not skewed, and positive
  when the distribution is positively skewed. Its
  calculation is based on the cubed deviations from
  the mean.
• Definition The skewness ratio is the value of
  the skewness statistic divided by its standard
  error. This measure is useful for determining the
  extent of skew. As a rule of thumb, when this
  ratio exceeds 2 in magnitude for small and
  moderate sized samples, the distribution is
  considered to be severely skewed.
• Useful when data are scale.
• To find the skewness ratio, click Analyze on the
  main menu bar, Descriptive Statistics, and then
  Explore. Click OK. Divide the skewness statistic
  by the standard error of the skew.

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Examples of Distributions of Different Shape
19
How the Shape of the Distribution Affects the
Mean and Median

• For a severely positively skewed distribution, in
  general, the mean is greater than the median.
• For a severely negatively skewed distribution, in
  general, the mean is less than the median.
• For a symmetric distribution, the mean equals the
  median.

20
Which Measure of Central Tendency Should One Use

• An article in the Wall Street Journal online
  (http//online.wsj.com/article/SB11879051854610711
  2.html) from August 24, 2007 reported the
  following
• The average cost of a wedding is between 27,400
  and 28,800.
• The median is approximately 15,000.
• How can we justify this apparent contradiction in
  the cost of a wedding?

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Applying What We have Learned

• What is the extent to which eighth-grade males
  expect larger incomes at age 30 than eighth-grade
  females?
• To what extent is there lack of consensus among
  males in their income expectations as compared to
  females?
• How are the answers to these questions influenced
  by the outliers and general shape of these
  distributions as shown in the boxplots in the
  last slide?

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Descriptive Statistics for Males and Females
23
Boxplots for Males and Females