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Probability & Statistics for P-8 Teachers. Chapter 2 Frequency Distributions and Graphs. What is Statistics?. Most people become familiar with probability and statistics through radio, television, newspapers, and magazines. For example, the following statements were found in newspapers.

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Probability statistics for p 8 teachers

Probability & Statistics for P-8 Teachers

Chapter 2

Frequency Distributions and Graphs


What is statistics
What is Statistics?

Most people become familiar with probability and statistics through radio, television, newspapers, and magazines. For example, the following statements were found in newspapers.

  • The U.S. is home to a quarter of the world’s cars

  • The mean NFL salary is $1.8 million in 2009

    • The league minimum is $295,000

    • The median salary is $770,000

  • The average cost of a wedding is $19,581

  • There are only thirteen blimps in the world.

  • Women who eat fish once a week are 29% less likely to develop heart disease


Various uses of statistics
Various uses of Statistics

Is Statistics important in our life?

  • Statistics is used in almost all fields of human endeavor.

  • In sports,

    • number of yards a running back gains during a football game, or the number of hits a baseball player gets in a season.

  • In public health

    • concerned with the number of residents who contract a new strain of flu virus during a certain year.

  • In education

    • are new methods of teaching are better than old ones?

  • Furthermore, statistics is used to analyze the results of surveys and as a tool in scientific research to make decisions based on controlled experiments.

  • Other uses of statistics include operations research, quality control, estimation, and prediction.


Statistics is the science of conducting studies to collect, organize, summarize, analyze, present, interpret and draw conclusions from data.

Any values (observations or measurements) that have been collected


What is statistics1
What Is Statistics?

  • Statistics is a collection of tools and methods, designed to help us understand the world.

  • Statistics are calculations and interpretations made from data.

  • Statistics helps us to understand the real, imperfect world in which we live.


What is statistics2
What Is Statistics?


Think show tell
Think, Show, Tell

  • There are three simple steps to doing Statistics right:

    first. Know where you’re headed and why.

    is about the mechanics of calculating statistics and graphical displays, which are important (but are not the most important part of Statistics).

    what you’ve learned. You must explain your results so that someone else can understand your conclusions.


Terminology
Terminology

  • A variable is a characteristic or attribute that can assume different values.

  • The values that a variable can assume are called data.

  • A population consists of all subjects (human or otherwise) that are studied.

  • A sample is a subset of the population.


Population vs sample
Population vs sample


Descriptive and inferential statistics
Descriptive and Inferential Statistics

  • Descriptive statistics consists of the collection, organization, summarization, and presentation of data.

  • Inferential statistics consists of generalizing from samples to populations, performing estimations and hypothesis tests, determining relationships among variables, and making predictions.


Definitions
Definitions

Variable:A characteristic that varies from one person or

thing to another.

Qualitative variable:A non-numerically valued variable.

Quantitative variable:A numerically valued variable.

Discrete variable:A quantitative variable whose

possible values can be listed.

Continuous variable:A quantitative variable whose

possible values form some interval of numbers.


A discrete variable is a quantitative variable that either has a finite number of possible values or a countable number of possible values. The term “countable” means the values result from counting such as 0, 1, 2, 3, and so on.

A continuous variable is a quantitative variable that has an infinite number of possible values it can take on and can be measured to any desired level of accuracy.


Organizing data
Organizing Data

  • Data collected in original form is called raw data.

  • A frequency distribution is the organization of raw data in table form, using classes and frequencies.

  • Qualitative data can be placed in categories and organized in categorical frequency distributions.


Categorical frequency distribution
Categorical Frequency Distribution

A random sample of twenty-five patients in a hospital were given a blood test to determine their blood type.

Raw Data:

Construct a frequency distribution for the data.


Categorical frequency distribution1
Categorical Frequency Distribution

Raw Data:

IIII

IIII II

IIII IIII

IIII

5

7

9

4

20

28

36

16


Relative frequency distribution
Relative Frequency Distribution

The percentage of a class is called the relative frequency of the class

A table that provides all classes and their relative frequencies is called a relative-frequency distribution. Note that the relative frequencies sum to 1 (100%).

IIII

IIII II

IIII IIII

IIII

5

7

9

4

20

28

36

16

5/25

7/25

9/25

4/25

100


Grouped frequency distribution
Grouped Frequency Distribution

  • Grouped frequency distributions are used when the range of the data is large.

  • The smallest and largest possible data values in a class are the lower and upper class limits. Class boundaries separate the classes.

  • To find a class boundary, average the upper class limit of one class and the lower class limit of the next class.


Grouped frequency distribution1
Grouped Frequency Distribution

  • Theclass width can be calculated by subtracting

    • successive lower class limits (or boundaries)

    • successive upper class limits (or boundaries)

    • upper and lower class boundaries

  • The class midpoint Xmcan be calculated by averaging

    • upper and lower class limits (or boundaries)


Grouped frequency distribution2
Grouped Frequency Distribution

Rules for Classes in Grouped Frequency Distributions

  • There should be 5-20 classes.

  • The class width should be an odd number.

  • The classes must be mutually exclusive.

  • The classes must be continuous.

  • The classes must be exhaustive.

  • The classes must be equal in width

    (except in open-ended distributions).


Grouped frequency distribution3
Grouped Frequency Distribution

112 100 127 120 134 118 105 110 109 112

110 118 117 116 118 122 114 114 105 109

107 112 114 115 118 117 118 122 106 110

116 108 110 121 113 120 119 111 104 111

120 113 120 117 105 110 118 112 114 114

The following data represent the record high temperatures for each of the 50 states. Construct a grouped frequency distribution for the data using 7 classes.


Constructing a grouped frequency distribution
Constructing a Grouped Frequency Distribution

STEP 1 Determine the classes.

Find the class width by dividing the range by the number of classes 7.

Range = High – Low

= 134 – 100 = 34

Width = Range/7 = 34/7 = 5

Rounding Rule: Always round up if a remainder.


Constructing a grouped frequency distribution1
Constructing a Grouped Frequency Distribution

  • For convenience sake, we will choose the lowest data value, 100, for the first lower class limit.

  • The subsequent lower class limits are found by adding the width to the previous lower class limits.

Class Limits

100 -

105 -

110 -

115 -

120 -

125 -

130 -

  • The first upper class limit is one less than the next lower class limit.

  • The subsequent upper class limits are found by adding the width to the previous upper class limits.

104

109

114

119

124

129

134


Constructing a grouped frequency distribution2
Constructing a Grouped Frequency Distribution

  • The class boundary is midway between an upper class limit and a subsequent lower class limit. 104,104.5,105

99.5 - 104.5

104.5 - 109.5

109.5 - 114.5

114.5 - 119.5

119.5 - 124.5

124.5 - 129.5

129.5 - 134.5


Constructing a grouped frequency distribution3
Constructing a Grouped Frequency Distribution

STEP 2 Tally the data.

STEP 3 Find the frequencies.

99.5 - 104.5

104.5 - 109.5

109.5 - 114.5

114.5 - 119.5

119.5 - 124.5

124.5 - 129.5

129.5 - 134.5

2

8

18

13

7

1

1


Cumulative frequency distribution
Cumulative Frequency Distribution

Sometimes it is helpful to keep a running total of the frequencies. This is called a cumulative frequency distribution

99.5 - 104.5

104.5 - 109.5

109.5 - 114.5

114.5 - 119.5

119.5 - 124.5

124.5 - 129.5

129.5 - 134.5

2

8

18

13

7

1

1

2

10

28

41

48

49

50


Data analysis
Data Analysis

  • There are three simple rules of data analysis:

    • Make a picture—things may be revealed that are not obvious in the raw data. These will be things to think about.

    • Make a picture—important features of and patterns in the data will show up. You may also see things that you did not expect.

    • Make a picture—the best way to tell others about your data is with a well-chosen picture.


Let s draw pictures
Let’s Draw Pictures

Most Common Graphs in Research

  • Histogram

  • Frequency Polygon

  • Cumulative Frequency Polygon (Ogive)

  • Bar Graph

  • Pareto Chart

  • Time-Series Graph

  • Pie Graph

  • Dot Plot

  • Stem and Leaf Plot

  • Scatter Plot


Histograms
Histograms

The histogram is a graph that displays the data by using vertical bars of various heights to represent the frequencies of the classes.

The class boundaries are represented on the horizontal axis.


Histograms1
Histograms

Construct a histogram to represent the data for the record high temperatures for each of the 50 states

Histograms use class boundaries and frequencies of the classes.

99.5 - 104.5

104.5 - 109.5

109.5 - 114.5

114.5 - 119.5

119.5 - 124.5

124.5 - 129.5

129.5 - 134.5

2

8

18

13

7

1

1


Histograms2
Histograms

Class boundaries represent the x-axis

Frequencies represent the y-axis


Frequency polygon
Frequency Polygon

  • The frequency polygon is a graph that displays the data by using lines that connect points plotted for the frequencies at the class midpoints. The frequencies are represented by the heights of the points.

  • The class midpoints are represented on the horizontal axis.


Frequency polygons
Frequency Polygons

Construct a frequency polygon to represent the data for the record high temperatures for each of the 50 states

Frequency polygons use class midpoints and frequencies of the classes.

102

107

112

117

122

127

132

2

8

18

13

7

1

1


Frequency polygons1
Frequency Polygons

A frequency polygon

is anchored on the

x-axis before the first

class and after the

last class.


Ogives cumulative frequency polygon
Ogives(Cumulative Frequency Polygon)

  • The ogiveis a graph that represents the cumulative frequencies for the classes in a frequency distribution.

  • The upper class boundaries are represented on the horizontal axis.


Ogives
Ogives

Construct an Ogive to represent the data for the record high temperatures for each of the 50 states

Ogives use upper class boundaries and cumulative frequencies of the classes.

99.5 - 104.5

104.5 - 109.5

109.5 - 114.5

114.5 - 119.5

119.5 - 124.5

124.5 - 129.5

129.5 - 134.5

2

10

28

41

48

49

50



Relative frequencies
Relative Frequencies

If proportions are used instead of frequencies, the graphs are called relative frequency graphs.

Relative frequency graphs are used when the proportion of data values that fall into a given class is more important than the actual number of data values that fall into that class.


Construct a histogram, frequency polygon, and ogive using relative frequencies for the

distribution (shown here) of the miles that 20 randomly selected runners ran during a

given week.


Graphs
Graphs relative frequencies for the


Shapes of distributions
Shapes of Distributions relative frequencies for the


Shapes of distributions1
Shapes of Distributions relative frequencies for the


Other types of graphs bar graphs
Other Types of Graphs relative frequencies for theBar Graphs


Other types of graphs pareto charts
Other Types of Graphs relative frequencies for thePareto Charts


Other types of graphs time series graphs
Other Types of Graphs relative frequencies for theTime Series Graphs


Other types of graphs pie graphs
Other Types of Graphs relative frequencies for thePie Graphs


Other types of graphs dot plots
Other Types of Graphs relative frequencies for theDot Plots


Other types of graphs stem and leaf plots
Other Types of Graphs relative frequencies for theStem and Leaf Plots

A stem and leaf plot is a data plot that uses part of a data value as the stem and part of the data value as the leaf to form groups or classes.

It has the advantage over grouped frequency distribution of retaining the actual data while showing them in graphic form.


Stem and leaf plots
Stem and Leaf Plots relative frequencies for the

At an outpatient testing center, the number of cardiograms performed each day for 20 days is shown. Construct a stem and leaf plot for the data.

25 31 20 32 13

14 43 2 57 23

36 32 33 32 44

32 52 44 51 45


Stem and leaf plots1
Stem and Leaf Plots relative frequencies for the

First, we list the leading digits of the numbers in the table(0, 1,. . ., 5)in a column, as shown to the left of the vertical rule. Next, we write the final digit of each number from the table to the right of the vertical rule in the row containing the appropriate leading digit.

25 31 20 32 13

14 43 2 57 23

36 32 33 32 44

32 52 44 51 45

Unordered Stem Plot

Ordered Stem Plot

0

2

1

3

4

2

5

0

3

3

1

2

6

2

3

2

2

4

3

4

4

5

5

7

2

1


Scatter plots and correlation
Scatter Plots and Correlation relative frequencies for the

  • A scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable x and the dependent variable y.

  • A scatter plot is used to determine if a relationship exists between the two variables.


Example motorcycle accidents
Example: Motorcycle Accidents relative frequencies for the

A researcher is interested in determining if there is a relationship between the number of motorcycle accidents and the number of motorcycle fatalities. The data are for a 10-year period. Draw a scatter plot for the data.

Step 1: Draw and label the x and y axes.

Step 2: Plot each point on the graph.


Example motorcycle accidents1
Example: Motorcycle Accidents relative frequencies for the


Analyzing the scatter plot
Analyzing the Scatter Plot relative frequencies for the

  • A positive linear relationship exists when the points fall approximately in an ascending straight line from left to right and both the x and y values increase at the same time.

  • A negative linear relationship exists when the points fall approximately in a descending straight line from left to right.

  • A nonlinear relationship exists when the points fall in a curved line.

  • It is said that no relationship exists when there is no discernable pattern of the points.


Analyzing the scatter plot1
Analyzing the Scatter Plot relative frequencies for the

(a) Positive linear relationship

(b) Negative linear relationship

(c) Nonlinear relationship

(d) No relationship


What can go wrong
What Can Go Wrong? relative frequencies for the

  • Don’t make a histogram of a categorical variable—bar charts or pie charts should be used for categorical data.


What can go wrong1
What Can Go Wrong? relative frequencies for the

  • Choose a bin width appropriate to the data.

    • Changing the bin width changes the appearance of the histogram:


What can go wrong2

Avoid inconsistent scales, either within the display or when comparing two displays.

Label clearly so a reader knows what the plot displays.

Good intentions, bad plot:

What Can Go Wrong?


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