chapter 2: frequency distributions - rutgers universitymmm431/quant_methods_f13/qm_lecture2.pdfon...

Displaying Data

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Displaying Data & Central Tendency

Frequency Distributions

• After collecting data, the first task for a researcher is to organize and summarize the data to get a general overview of the results. – Remember, this is the goal of descriptive statistical techniques.

• One method for simplifying and organizing data is to construct a frequency distribution.

– Frequency – describes the number of times or how often a category,

score, or range of scores occurs

– Frequency distribution – a summary display for a distribution of data

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Frequency Distribution Tables

• A simple frequency distribution table consists of two columns -

one listing categories on the scale of measurement (x) and

another for frequency (f).

– In the x column, values are listed in order from lowest to highest (or

from highest to lowest)

– For the frequency column, tallies are determined for each value (how

often each x value occurs in the data set). These tallies are the

frequencies for each x value.

– The sum of the frequencies should equal N.

• Frequency distributions can be computed for grouped or

ungrouped data

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Regular (ungrouped) Frequency Distribution

• When a frequency distribution table lists all of the individual

categories (x values) it is called a regular frequency

distribution.

x f

0 8

1 8

2 15

3 8

4 1

N=40

Example: x = number of naps toddlers take per day

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Grouped Frequency Distribution

• Sometimes, especially when dealing with continuous

variables, a set of scores covers a wide range of values

– In these situations, a list of all the x values would be too long to allow a

simple presentation of the data.

• In such cases, a grouped frequency distribution table is

used.

– In a grouped table, the x column lists groups of scores, called class

intervals, rather than individual values.

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34 12 33 11

16 31 17 72

14 17 18 13

17 6 17 24

56 9 10 18

7 10 67 17

83 77 5 9

16 18 28 35

16 30 70 18

15 10 13 12

5 6 7 9

9 10 10 10

11 12 12 13

13 14 15 16

16 16 17 17

17 17 17 18

18 18 18 24

28 30 31 33

34 35 56 67

70 72 77 83

Example: x = college course enrollment

Sorted values:

x f

0-10 5

10-20 22

20-30 2

30-40 5

40-50 0

50-60 1

60-70 1

70-80 3

80-90 1

N=40

Grouped frequency distribution

Note: I prefer to use real limits when specifying

intervals. Your book uses apparent limits. You

can use either.

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Grouped Frequency Distributions: Guidelines

• Sort your data first, it makes building the frequency

distributions easier

• Decide on interval width and number of intervals

– You should have about 5-20 intervals

– All intervals should have the same width

– Your interval width should be a relatively simple number

• Examples: 10, 5, 2, 1, 0.5

– Your set of intervals should cover all observed values and should not

overlap

• I.e., no individual score should fall in more than one interval

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Relative Frequencies & Percentages

• Often, researchers are more interested in the relative frequency (or proportion) of individuals in each category than in the total number. – Remember from the last lecture that we usually measure statistics on

samples to infer parameters of populations

– The relative frequency of a sample approximates the relative frequency of the population, whereas the raw frequency of a sample does not.

• The relative frequency distribution table lists the proportion (p) for each category: p = f/N. The sum of the p column should equal 1.00. – Alternatively, the table could list the percentage of the distribution

corresponding to each X value. The percentage is found by multiplying p by 100. The sum of the percentage column should equal 100%.

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Relative Frequencies & Percentages

x f p (or f/N) %

0-10 5 0.125 12.5

10-20 22 0.550 55.0

20-30 2 0.050 5.0

30-40 5 0.125 12.5

40-50 0 0.000 0.0

50-60 1 0.025 2.5

60-70 1 0.025 2.5

70-80 3 0.075 7.5

80-90 1 0.025 2.5

Total 40 1 100%

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Cumulative Frequencies, Proportions, &

Percentages

• Cumulative frequencies, proportions, or percentages

describe the sum of frequencies, proportions, or percentages

across a series of intervals

• Usually refers to bottom-up sum of frequencies

– E.g., the number of college courses with at least k students

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Cumulative Frequencies & Percentages

x f Cumulative

Freq. %

Cumulative %

0-10 5 5 12.5% 12.5%

10-20 22 27 55.0% 67.5%

20-30 2 29 5.0% 72.5%

30-40 5 34 12.5% 85.0%

40-50 0 34 0.0% 85.0%

50-60 1 35 2.5% 87.5%

60-70 1 36 2.5% 90.0%

70-80 3 39 7.5% 97.5%

80-90 1 40 2.5% 100.0%

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Frequency Distribution Graphs

• In a frequency distribution graph, the score categories (X

values) are listed on the X axis and the frequencies are listed

on the Y axis.

• When the score categories consist of numerical scores from

an interval or ratio scale, the graph should be either a

histogram or a polygon.

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Bar Plots & Histograms

• Bar plots are plots showing the relationship between two

variables. Usually, the height of a bar represents the value of

a dependent variable when the independent variable consists

of nominal or ordinal category labels.

• Histograms are bar plots in which the rectangles are

centered above each score (or class interval) and the heights

of the bars correspond to the frequencies (or relative

frequencies) of the scores.

– The widths of bars should extend to the real limits of the class

intervals, so that adjacent bars touch.

Note: Proper histograms actually represent frequencies in terms of the area rather than

the height of bars, but we won’t worry about that distinction in this course

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x f

brown 14

red 14

blue 10

orange 7

green 6

yellow 5

n=56

Bar Plot Example: M&Ms Colors

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x f

0-10 5

10-20 22

20-30 2

30-40 5

40-50 0

50-60 1

60-70 1

70-80 3

80-90 1

N=40

Histogram Example: Course Enrollment

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Line Plots & Frequency Polygons

• Line plots are plots in which dots (rather than rectangles) are

centered above one score in each of a pair of scores, with the

height of the dot determined by the second score, and lines

are drawn to connect the dots.

• A frequency polygon is a type of line plot analogous to a

histogram, where the heights of the dots correspond to

frequencies or relative frequencies of scores or intervals.

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Line Plot Example: Monthly Avg. Temps in NJ

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Frequency Polygons: Example

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Scatter Plots

A scatter plot (or scatter gram) displays discrete data points (x, y)

to summarize the relationship between two variables

Height Weight

70 150

67 140

72 180

75 190

68 145

69 150

71.5 164

71 140

72 142

69 136

67 123

68 155

66 140

72 145

73.5 160

73 190

69 155

73 165

72 150

74 190

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Theoretical Distributions, Probability Densities

& Smooth Curves

• If the scores in the population are continuous variables, then

the theoretical distributions describing them will often be

depicted as smooth curves

– Examples of this include the normal distribution (i.e., “the bell curve”) as

well as most of the test statistic distributions that we will deal with in this

course (e.g., the t distribution, the F distribution, the chi-square

distribution)

• The smooth curves represent the expectation that in a large

population, relative frequencies should change smoothly as a

function of a continuous variable.

– These smooth curves actually represent probability densities, which are

related to relative frequencies

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Frequency & Probability Distribution Graphs

• Frequency & probability distribution graphs are useful because they show the entire set of scores.

• At a glance, you can determine the highest score, the lowest score, and where the scores are centered.

• The graph also shows whether the scores are clustered together or scattered over a wide range.

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Distribution Shape

• A graph shows the shape of the distribution.

• A distribution is symmetrical if the left side of the graph is (roughly) a mirror image of the right side.

• One example of a symmetrical distribution is the bell-shaped normal distribution.

• On the other hand, distributions are skewed when scores pile up on one side of the distribution, leaving a "tail" of a few extreme values on the other side.

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Distribution Shape

• In a positively skewed distribution, the scores tend to pile up

on the left side of the distribution with the tail tapering off to

the right.

• In a negatively skewed distribution, the scores tend to pile

up on the right side and the tail points to the left.

• A unimodal distribution has one peak

• A bimodal (multimodal) distribution has two (multiple) peaks

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Central Tendency

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Central Tendency

• In general terms, central tendency is a statistical measure

that determines a single value that accurately describes the

center of the distribution and represents the entire distribution

of scores.

• The goal of central tendency is to identify the single value that

is the best representative for the entire set of data.

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Central Tendency

• By identifying the "average score," central tendency allows researchers to summarize or condense a large set of data into a single value.

• Thus, central tendency serves as a descriptive statistic because it allows researchers to describe or present a set of data in a very simplified, concise form.

• In addition, it is possible to compare two (or more) sets of data by simply comparing the average score (central tendency) for one set versus the average score for another set.

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The Mean, the Median, and the Mode

• No single procedure always produces a good, representative

value. Therefore, researchers have developed three

commonly used techniques for measuring central tendency:

the mean, the median, and the mode.

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The Mean

• The mean is the most commonly used measure of central tendency. – The population mean is denoted by:

– The sample mean is denoted by: M or

• Computation of the mean requires scores that are numerical values measured on an interval or ratio scale.

• The mean is obtained by computing the sum, or total, for the entire set of scores, then dividing this sum by the number of scores.

1x

N

X

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Changing the Mean

• Because the calculation of the mean involves every score in the distribution, changing the value of any score will change the value of the mean.

• Modifying a distribution by discarding scores or by adding new scores will usually change the value of the mean.

• To determine how the mean will be affected for any specific situation you must consider: 1) how the number of scores is affected, and 2) how the sum of the scores is affected.

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Changing the Mean

• If a constant value is added to every score in a distribution,

then the same constant value is added to the mean. Also, if

every score is multiplied by a constant value, then the mean is

also multiplied by the same constant value.

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The Weighted Mean

• When combining data from samples with different sizes, you

can compute the combined mean from the sample means

using the following formula:

• For example, consider the following samples:

– Sample 1: x = {6,2,6,8,3}; M = 5.0; n = 5

– Sample 2: x = {3,6,13,4}; M = 6.5; n = 4

– Sample 3: x = {3,4,2}; M = 3.0; n = 3

wh1

, ere W nMN

M N n

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When the Mean Won’t Work

• Although the mean is the most commonly used measure of central tendency, there are situations where the mean does not provide a good, representative value, or where you cannot compute a mean at all.

• When a distribution contains a few extreme scores (or is very skewed), the

mean will be pulled toward the extremes. In these cases, the mean will not provide a "central" value.

• With data from a nominal scale it is impossible to compute a mean, and when data are measured on an ordinal scale (ranks), it is usually inappropriate to compute a mean.

• Thus, the mean does not always work as a measure of central tendency and it is necessary to have alternative procedures available.

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The Median

• If the scores in a distribution are listed in order from smallest

to largest, the median is defined as the midpoint of the list.

• This means that computation of the median requires scores that can be

placed in rank order (i.e., ordinal, interval, or ratio)

• The median divides the scores so that 50% of the scores in the

distribution have values that are equal to or less than the median.

• Usually, the median can be found by a simple counting

procedure:

1. With an odd number of scores, list the values in order, and the median

is the middle score in the list.

2. With an even number of scores, list the values in order, and the

median is half-way between the middle two scores.

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The Median

• One advantage of the median is that it is relatively unaffected

by extreme scores.

• Thus, the median tends to stay in the "center" of the

distribution even when there are a few extreme scores or

when the distribution is very skewed.

• In these situations, the median serves as a good alternative to

the mean.

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The Mode

• The mode is defined as the most frequently occurring

category or score in the distribution.

• In a frequency distribution graph, the mode is the category or

score corresponding to the peak or high point of the

distribution.

• The mode can be determined for data measured on any scale

of measurement: nominal, ordinal, interval, or ratio.

– The mode is the only measure of central tendency that can be used for

data measured on a nominal scale.

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Bimodal Distributions

• It is possible for a distribution to have more than one mode.

Such a distribution is called bimodal. (Note that a distribution

can have only one mean and only one median.)

• In addition, the term "mode" is often used to describe a peak

in a distribution that is not really the highest point. Thus, a

distribution may have a major mode at the highest peak and a

minor mode at a secondary peak in a different location.

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Central Tendency and the Shape of the Distribution

• Because the mean, the median, and the mode are all

measuring central tendency, the three measures are often

systematically related to each other.

• In a symmetrical distribution, for example, the mean and

median will always be equal.

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Central Tendency and the Shape of the Distribution

• If a symmetrical distribution has only one mode, the mode,

mean, and median will all have the same value.

• In a skewed distribution, the mode will be located at the peak

on one side and the mean usually will be displaced toward the

tail on the other side.

• The median is usually located between the mean and the

mode.

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Central Tendency and the Shape of the

Distribution

chapter 2: frequency distributions - rutgers universitymmm431/quant_methods_f13/qm_lecture2.pdfon...

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