sample vs. population - weebly

Sample vs. Population

Population Sample

Descriptive Statistics

• Organize

• Summarize

• Simplify

• Presentation of

data

Describing data


Descriptive Statistics are Used by Researchers

to Report on Populations and Samples


Types of descriptive statistics:

Organize Data

Tables

Graphs

Summarize Data

Central Tendency

Variation


Types of descriptive statistics:

Organize Data

Tables

Frequency Distributions

Relative Frequency Distributions

Graphs

Bar Chart or Histogram

6


After collecting data, the first task for a researcheris to organize and simplify the data so that it ispossible to get a general overview of the results.

This is the goal of descriptive statistical techniques.

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

7


A frequency distribution is an organized

tabulation showing exactly how many individuals

are located in each category on the scale of

measurement. A frequency distribution presents

an organized picture of the entire set of scores,

and it shows where each individual is located

relative to others in the distribution.

8

Frequency Distribution Tables

A frequency distribution table consists of at least two columns - one listing categories on the scale of measurement (X) and another for frequency (f).

In the X column, values are listed from the highest to lowest, without skipping any.

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.

9

Frequency Distribution Tables

A third column can be used for the proportion (p) for each category: p = f/N. The sum of the p column should equal 1.00.

A fourth column can display 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 is 100%.

A frequency distribution is a tabular summary ofdata showing the frequency (or number) of itemsin each of several non overlapping classes.

The objective is to provide insights about the datathat cannot be quickly obtained by looking only atthe original data.

Frequency Distribution

Example: Scores of students from

petroleum engineering department

Scores of 20 students are listed below

Below Average

Above Average

Above Average

Average

Above Average

Average

Above Average

Average

Above Average

Below Average

Low

Excellent

Above Average

Average

Above Average

Above Average

Below Average

Low

Above Average

Average

Average


Low

Below Average

Average

Above Average

Excellent

2

3

5

9

1

Total 20

Scores Frequency

The relative frequency of a class is the fraction orproportion of the total number of data itemsbelonging to the class.

A relative frequency distribution is a tabularsummary of a set of data showing the relativefrequency for each class.

Relative Frequency Distribution

Percent Frequency Distribution

The percent frequency of a class is the relativefrequency multiplied by 100.

A percent frequency distribution is a tabularsummary of a set of data showing the percentfrequency for each class.

Relative Frequency andPercent Frequency Distributions

Low

Below Average

Average

Above Average

Excellent

.10

.15

.25

.45

.05

Total 1.00

10

15

25

45

5

100

Relative

Frequency

Percent

FrequencyScores

.10(100) = 10

1/20 = .05

16

Frequency distribution graphs

Frequency 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.

17

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.

Bar Graph

A bar graph is a graphical device for depictingqualitative data.

On one axis (usually the horizontal axis), we specifythe labels that are used for each of the classes.

A frequency, relative frequency, or percent frequencyscale can be used for the other axis (usually thevertical axis).

Using a bar of fixed width drawn above each classlabel, we extend the height appropriately.

The bars are separated to emphasize the fact that eachclass is a separate category.

Low BelowAverage

Average AboveAverage

Excellent

Fre

qu

en

cy

Rating

Bar Graph

1

2

3

4

5

6

7

8

9

10Scores of students

Pie Chart

The pie chart is a commonly used graphical devicefor presenting relative frequency distributions forqualitative data.

First draw a circle; then use the relative

frequencies to subdivide the circle

into sectors that correspond to the

relative frequency for each class.

Since there are 360 degrees in a circle,

a class with a relative frequency of .25 would

consume .25(360) = 90 degrees of the circle.

BelowAverage

15%

Average25%

AboveAverage

45%

Low10%

Excellent5%

Scores of students

Pie Chart

Grouped Frequency Distribution

23


Sometimes, however, a set of scores covers a

wide range of values. In these situations, a list of

all the X values would be quite long - too long to

be a “simple” presentation of the data.

To remedy this situation, a grouped frequency

distribution table is used.

24


In a grouped table, the X column lists groups of

scores, called class intervals, rather than individual

values.

These intervals all have the same width, usually a

simple number such as 2, 5, 10, and so on.

Each interval begins with a value that is a multiple of

the interval width. The interval width is selected so

that the table will have approximately ten intervals.

Example: The following Table shows the permeability values that

were taken from 50 wells in a case study.

Sample of permeability values for 50 Samples

91 78 93 57 75 52 99 80 97 62

71 69 72 89 66 75 79 75 72 76

104 74 62 68 97 105 77 65 80 109

85 97 88 68 83 68 71 69 67 74

62 82 98 101 79 105 79 69 62 73

Including a line in the table for every

Sample is not a good idea.

Need to categorize.


Guidelines for Selecting Number of

Classes

• Use between 5 and 20 classes.

• Data sets with a larger number of elementsusually require a larger number of classes.

• Smaller data sets usually require fewer classes


Guidelines for Selecting Width of Classes

Largest Data Value Smallest Data Value

Number of Classes

•Use classes of equal width.

•Approximate Class Width =

Summarizing Quantitative Data


Relative Frequency and Percent


Histogram

Cumulative Distributions


For permeability values , if we choose six

classes:

50-59

60-69

70-79

80-89

90-99

100-109

2

13

16

7

7

5

Total 50

permeability Frequency

Approximate Class Width = (109 - 52)/6 = 9.5 10

Relative Frequency and Percent Frequency

Distributions

50-59

60-69

70-79

80-89

90-99

100-109

permeability

.04

.26

.32

.14

.14

.10

Total 1.00

Relative

Frequency

4

26

32

14

14

10

100

Percent

Frequency

2/50 .04(100)

Pre

vie

w c

um

ula

tive

freq

uen

cies

her

e.

• Only 4% of the permeability values are in the 50-59 class.

• The greatest percentage (32% or almost one-third)of the permeability values are in the 70-79 class.

• 30% of the permeability values are under 70.

• 10% of the permeability values are 100 or more.

Insights Gained from the Percent Frequency

Distribution

Relative Frequency and

Percent Frequency Distributions

32

Relative frequency

Many populations are so large that it is

impossible to know the exact number of

individuals (frequency) for any specific

category.

In these situations, population distributions

can be shown using relative frequency

instead of the absolute number of individuals

for each category.

2007©BOLD Educational Software

Ages f Relative Freq.

10 up to 19 2 2/6020 up to 29 1 1/6030 up to 39 5

40 up to 49 20

50 up to 59 25

60 up to 69 3

70 up to 79 4

Total 60 1

Practice: Determine the Relative Frequency Distribution

34

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.

35

Histograms

In a histogram, a bar is centered above each

score (or class interval) so that the height of

the bar corresponds to the frequency and the

width extends to the real limits, so that

adjacent bars touch.

Histogram

Another common graphical presentation ofquantitative data is a histogram.

The variable of interest is placed on the horizontalaxis.

A rectangle is drawn above each class interval withits height corresponding to the interval’s frequency,relative frequency, or percent frequency.

Unlike a bar graph, a histogram has no naturalseparation between rectangles of adjacent classes.

In informal discussions bar graphs and histograms

are often equated. In this class you should be

careful to keep them straight.

Histogram

2

4

6

8

10

12

14

16

18

permeability

Fre

qu

en

cy

5059 6069 7079 8089 9099 100-110

permeability values

39

Frequency distribution graphs

Frequency 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.

Cumulative frequency distribution shows thenumber of items with values less than or equal tothe upper limit of each class..

Cumulative relative frequency distribution – showsthe proportion of items with values less than orequal to the upper limit of each class.


Cumulative percent frequency distribution – showsthe percentage of items with values less than orequal to the upper limit of each class.


permeability values

< 59

< 69

< 79

< 89

< 99

< 109

permeabilityCumulativeFrequency

CumulativeRelative

Frequency

CumulativePercent

Frequency

2

15

31

38

45

50

.04

.30

.62

.76

.90

1.00

4

30

62

76

90

100

2 + 13 15/50 .30(100)

permeability

20

40

60

80

100

Cu

mu

lati

ve

Per

cen

t F

req

uen

cy

50 60 70 80 90 100 110

(89.5, 76)

Cumulative Percent Frequencies

permeability values

Data analysis using SPSS

(Statistical package for social science)


Class A--IQs of 13 Students

102 115

128 109

131 89

98 106

140 119

93 97

110

Class B--IQs of 13 Students

127 162

131 103

96 111

80 109

93 87

120 105

109

An Illustration:

Which Group is Smarter?

Each individual may be different. If you try to understand a group by remembering the qualities of each member, you become overwhelmed and fail to understand the group.

SPSS Output for Frequency Distribution

IQ

1 4.2 4.2 4.2

1 4.2 4.2 8.3

1 4.2 4.2 12.5

2 8.3 8.3 20.8

1 4.2 4.2 25.0

1 4.2 4.2 29.2

1 4.2 4.2 33.3

1 4.2 4.2 37.5

1 4.2 4.2 41.7

1 4.2 4.2 45.8

1 4.2 4.2 50.0

1 4.2 4.2 54.2

1 4.2 4.2 58.3

1 4.2 4.2 62.5

1 4.2 4.2 66.7

1 4.2 4.2 70.8

1 4.2 4.2 75.0

1 4.2 4.2 79.2

1 4.2 4.2 83.3

2 8.3 8.3 91.7

1 4.2 4.2 95.8

1 4.2 4.2 100.0

24 100.0 100.0

82.00

87.00

89.00

93.00

96.00

97.00

98.00

102.00

103.00

105.00

106.00

107.00

109.00

111.00

115.00

119.00

120.00

127.00

128.00

131.00

140.00

162.00

Total

Valid

Frequency Percent Valid Percent

Cumulative

Percent


Frequency Distribution of IQ for Two Classes

IQ Frequency

82.00 1

87.00 1

89.00 1

93.00 2

96.00 1

97.00 1

98.00 1

102.00 1

103.00 1

105.00 1

106.00 1

107.00 1

109.00 1

111.00 1

115.00 1

119.00 1

120.00 1

127.00 1

128.00 1

131.00 2

140.00 1

162.00 1

Total 24

Relative Frequency Distribution

Relative Frequency Distribution of IQ for Two Classes

IQ Frequency Percent Valid Percent Cumulative Percent

82.00 1 4.2 4.2 4.2

87.00 1 4.2 4.2 8.3

89.00 1 4.2 4.2 12.5

93.00 2 8.3 8.3 20.8

96.00 1 4.2 4.2 25.0

97.00 1 4.2 4.2 29.2

98.00 1 4.2 4.2 33.3

102.00 1 4.2 4.2 37.5

103.00 1 4.2 4.2 41.7

105.00 1 4.2 4.2 45.8

106.00 1 4.2 4.2 50.0

107.00 1 4.2 4.2 54.2

109.00 1 4.2 4.2 58.3

111.00 1 4.2 4.2 62.5

115.00 1 4.2 4.2 66.7

119.00 1 4.2 4.2 70.8

120.00 1 4.2 4.2 75.0

127.00 1 4.2 4.2 79.2

128.00 1 4.2 4.2 83.3

131.00 2 8.3 8.3 91.7

140.00 1 4.2 4.2 95.8

162.00 1 4.2 4.2 100.0

Total 24 100.0 100.0

Grouped Relative Frequency Distribution

Relative Frequency Distribution of IQ for Two Classes

IQ FrequencyPercent Cumulative Percent

80 – 89 3 12.5 12.5

90 – 99 5 20.8 33.3

100 – 109 6 25.0 58.3

110 – 119 3 12.5 70.8

120 – 129 3 12.5 83.3

130 – 139 2 8.3 91.6

140 – 149 1 4.2 95.8

150 and over 1 4.2 100.0

Total 24 100.0 100.0

SPSS Output for Histogram

80.00 100.00 120.00 140.00 160.00

IQ

0

1

2

3

4

5

6F

req

ue

nc

y

Mean = 110.4583Std. Dev. = 19.00338N = 24

Histogram

80.00 100.00 120.00 140.00 160.00

IQ

0

1

2

3

4

5

6

Fre

qu

en

cy

Histogram of IQ Scores for Two Classes

Bar Graph

1.00 2.00

Class

0

2

4

6

8

10

12

Co

un

t

Bar Graph of Number of Students in Two Classes

sample vs. population - weebly

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