Hybrid Statistics Frequency Distributions and Their...

1 | P a g e Hannah Province – Mathematics Department Southwest Tennessee Community College

Hybrid Statistics – Chapter 2

Section 2.1 – Frequency Distributions and Their Graphs

Objectives: Construct frequency distributions

Construct frequency histograms

Frequency polygons

Relative frequency histograms

Ogives

Frequency Distribution

A frequency table partitions data into classes or intervals and shows how many data values are in each

class. The classes or intervals are constructed so that each data falls into exactly one class. If the

frequency is converted into percentage of individuals then we have a relative frequency table.

Steps to Constructing a Frequency Distribution

1. Decide on the number of classes.

a. Usually between 5 and 20; otherwise, it may be difficult to detect any patterns. The

problem will usually tell you how many classes.

2. Find the class width.

a. Determine the range of the data. (Highest Obs. – Lowest Obs.)

b. Divide the range by the number of classes.

c. Round up to the next convenient number.

3. Find the class limits.

a. You can use the minimum data entry as the lower limit of the first class.

b. Find the remaining lower limits (add the class width to the lower limit of the preceding

class).

c. Find the upper limit of the first class. Remember that classes cannot overlap.

d. Find the remaining upper class limits.

4. Count each data entry in the row of the appropriate class to find the total frequency, f, for each

class.


Determining the Class Limits (Upper and Lower)

Determining the Midpoint (In section 2.3 the Midpoint

will be denoted by the letter x)

Midpoint of a class

Determining the Relative Frequency

Relative Frequency of a class

Portion or percentage of the data that falls

in a particular class.

Determining the Cumulative Frequency

Cumulative frequency of a class

The sum of the frequency for that class and

all previous classes.

(Lower class limit) (Upper class limit)

2

59 11486.5

2

115 170142.5

2

171 226198.5

2

Class width = 56

50.17

30

80.27

30

60.2

30

+

+

6

1

31

9

Lower Class Limit Upper Class Limit


Example: Constructing a Frequency Distribution

The following sample data set lists the prices )in dollars) of 30 portable global positioning system (GPS)

navigators. Construct a frequency distribution that has seven classes.

90 130 400 200 350 70 325 250 150 250

275 270 150 130 59 200 160 450 300 130

220 100 200 400 200 250 95 180 170 150

Solution:


Graphs of Frequency Distributions

Frequency Histogram

A bar graph that represents the frequency distribution.

The horizontal scale is quantitative and measures the data values.

The vertical scale measures the frequencies of the classes.

Consecutive bars must touch.

Class boundaries

The numbers that separate classes without forming gaps between them.

The distance from the upper limit of the first class to the lower limit of the second class is

115 – 114 = 1. Half this distance is 0.5.

First class lower boundary = 59 – 0.5 = 58.5

First class upper boundary = 114 + 0.5 = 114.5

Class Boundaries

Class Boundaries

freq

uen

cy


Frequency Polygon

A line graph that emphasizes the continuous change in

frequencies.

Relative Frequency Histogram

Has the same shape and the same horizontal scale as the

corresponding frequency histogram.

The vertical scale measures the relative frequencies, not frequencies.

Cumulative Frequency Graph or Ogive

A line graph that displays the cumulative frequency of each

class at its upper class boundary.

The upper boundaries are marked on the horizontal axis.

The cumulative frequencies are marked on the vertical axis.

Constructing an Ogive

1. Construct a frequency distribution that includes cumulative frequencies as one

of the columns.

2. Specify the horizontal and vertical scales.

a. The horizontal scale consists of the upper class boundaries.

b. The vertical scale measures cumulative frequencies.

3. Plot points that represent the upper class boundaries and their corresponding

cumulative frequencies

4. Connect the points in order from left to right

5. The graph should start at the lower boundary of the first class (cumulative

frequency is zero) and should end at the upper boundary of the last class

(cumulative frequency is equal to the sample size).

Mid points

values

freq

uen

cy

Class Boundaries

values

rela

tive

freq

uen

cy

Class Boundaries

valuescu

mu

lati

ve

freq

uen

cy


Example: Frequency Histogram

Construct a frequency histogram for the Global Positioning system (GPS) navigators.

Solution: Frequency Histogram (using Midpoints). The frequencies go on the vertical axis in a nice index.

The class boundaries go on the horizontal axis.

You can see that more than half of the GPS navigators are priced below $226.50.


Example: Frequency Polygon

Construct a frequency polygon for the GPS navigators frequency distribution.

Solution: The graph should begin and end on the horizontal axis, so extend the left side to one class

width before the first class midpoint (86.5) and extend the right side to one class width after the last

class midpoint (422.5).

To get the value of 30.5 take 86.5 – 56(class width)

To get the value of 478.5 take 422.5 + 56(class width)

You can see that the frequency of GPS

navigators increases up to $142.50

and then decreases.


Example: Relative Frequency Histogram

Construct a relative frequency histogram for the GPS navigators frequency distribution

Solution: The relative frequencies go on the vertical axis in a nice index. The class boundaries go on the

horizontal axis.

From this graph you can see that 20%

of GPS navigators are priced between

$114.50 and $170.50.


Example: Ogive

Construct an ogive for the GPS navigators frequency distribution.

Solution: The cumulative frequencies go on the vertical axis in a nice index. The class boundaries go on

the horizontal axis.

From the ogive, you can see that

about 25 GPS navigators cost $300 or

less. The greatest increase occurs

between $114.50 and $170.50.


Section 2.2 – More Graphs and Displays

Objectives: Graph quantitative data using stem-and-leaf plots and dot plots

Graph qualitative data using pie charts and Pareto charts

Graph paired data sets using scatter plots and time series charts

Graphing Quantitative Data Sets

Stem-and-leaf plot

Each number is separated into a stem and a leaf.

Similar to a histogram.

Still contains original data values.

Dot plot

Each data entry is plotted, using a point, above a horizontal axis

Pie Chart

A circle is divided into sectors that represent categories.

The area of each sector is proportional to the frequency of each category.

Pareto Chart

A vertical bar graph in which the height of each bar represents frequency or relative frequency.

The bars are positioned in order of decreasing height, with the tallest bar positioned at the left.

Paired Data Sets

Each entry in one data set corresponds to one entry in a second data set.

Graph using a scatter plot.

The ordered pairs are graphed as points in a coordinate plane. Used to show the relationship

between two quantitative variables.

Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45

26

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45

x

y


Time Series

Data set is composed of quantitative entries taken at regular intervals over a period of time.

e.g., the amount of precipitation measured each day for one month.

Use a time series chart to graph.

Example: Constructing a Stem-and-Leaf Plot

The following are the numbers of text messages sent last month by the cellular phone users on one floor

of a college dormitory. Display the data in a stem-and-leaf plot.

Solution: The data entries go from a low of 78 to a high of 159.

Use the rightmost digit as the leaf. For instance, 78 = 7 | 8 and 159 = 15 | 9

List the stems, 7 to 15, to the left of a vertical line.

For each data entry, list a leaf to the right of its stem. Do not skip numbers!

From the display, you can conclude that more than 50% of the cellular phone users sent between 110

and 130 text messages.

155 159 144 129 105 145 126 116 130 114 122 112 112 142 126 118 108 122 121 109 140 126 119 113 117 118 109 109 119 139 122 78 133 126 123 145 121 134 124 119 132 133 124 129 112 126 148 147

Include a key to

identify the values of

the data.

time

Quanti

tati

ve

dat

a


Example: Constructing a Dot Plot

Use a dot plot organize the text messaging data.

So that each data entry is included in the dot plot, the horizontal axis should include numbers between

70 and 160.

To represent a data entry, plot a point above the entry's position on the axis.

If an entry is repeated, plot another point above the previous point.

Solution: From the dot plot, you can see that most values cluster between 105 and 148 and the value

that occurs the most is 126. You can also see that 78 is an unusual data value.

155 159 144 129 105 145 126 116 130 114 122 112 112 142 126 118 108 122 121 109 140 126 119 113 117 118 109 109 119 139 122 78 133 126 123 145 121 134 124 119 132 133 124 129 112 126 148 147


Example: Constructing a Pie Chart

The numbers of earned degrees conferred (in thousands) in 2007 are shown in the table. Use a pie chart

to organize the data. (Source: U.S. National Center for Educational Statistics)

Solution:


Section 2.3 - Measures of Central Tendency

Objectives

Determine the mean, median, and mode of a population and of a sample

Determine the weighted mean of a data set and the mean of a frequency distribution

Describe the shape of a distribution as symmetric, uniform, or skewed and compare the mean

and median for each

Measure of central tendency

A value that represents a typical, or central, entry of a data set.

Most common measures of central tendency:

Mean

Median

Mode

Mean (average)

The sum of all the data entries divided by the number of entries.

Sigma notation: Σx = add all of the data entries (x) in the data set.

Population mean: Sample mean:

Median

The value that lies in the middle of the data when the data set is ordered.

Measures the center of an ordered data set by dividing it into two equal parts.

If the data set has an odd number of entries: median is the middle data entry. even number

of entries: median is the mean of the two middle data entries.

The median is resistant to outliers

Finding the median:

1). Order the data from smallest to largest.

2). For an odd number of data values:

Median = Middle data value

3). For an even number of data values:

Mode

The data entry that occurs with the greatest frequency.

If no entry is repeated the data set has no mode.

If two entries occur with the same greatest frequency, each entry is a mode (bimodal).

Note: not every data set has a mode

i.e. data: 10 10 10 10 has no mode.

x

N

xx

n

Sum of middle two valuesMedian

2


Example: Find the mean, median and mode of the following data set.

{3, 8, 5, 4, 8, 4, 10}

Solution:

Comparing the Mean, Median, and Mode

• All three measures describe a typical entry of a data set.

• Advantage of using the mean:

The mean is a reliable measure because it takes into account every entry of a data set.

• Disadvantage of using the mean:

Greatly affected by outliers (a data entry that is far removed from the other entries in

the data set).

Outliers in a data set are data values that are very different from other measurements in the data set.

They many indicate that an error occurred or the data may be an actual data point.

Do you include Outliers in statistical analysis?

It depends, any decision about outliers should include people what are familiar with the field

and the purpose of the study.

Weighted Mean

• The mean of a data set whose entries have varying weights.

• where w is the weight of each entry x

Example: Grades: Given that the exam average is 85, the final exam score was 87, the homework

average was 98. Find the weighted average if the tests are worth 40% and the final was worth 40% and

the homework was 20%.

Solution:

( )x wx

w

How to With your Calculator:

Press STAT ⟶ EDIT

This is where you enter your data


Mean of a Frequency Distribution

• Approximated by

where x is the midpoints and f is the frequencies of a class, respectively

Finding the Mean of a Frequency Distribution

Example: Find the Mean of a Frequency Distribution – Grouped Data

Use the frequency distribution to approximate the mean number of minutes that a sample of Internet

subscribers spent online during their most recent session.

( )x fx n f

n

In Words In Symbols

( )x fx

n

(lower limit)+(upper limit)

2x

( )x f

n f

1. Find the midpoint of each class.

2. Find the sum of the products of the midpoints and the frequencies.

3. Find the sum of the frequencies.

4. Find the mean of the frequency distribution.

Solution:


Distribution Shapes

Symmetric Distribution

A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves

are approximately mirror images.

• Mound-shaped symmetrical – mean = median = mode.

Uniform Distribution (rectangular)

All entries or classes in the distribution have equal or approximately equal frequencies.

Uniform-shaped – mean = median. There would be no mode here.


Skewed Left Distribution (negatively skewed)

The “tail” of the graph elongates more to the left. The mean is to the left of the median.

• Left Skewed shaped – mean < median < mode.

Skewed Right Distribution (positively skewed)

The “tail” of the graph elongates more to the right. The mean is to the right of the median.

• Right Skewed shaped – mean > median > mode.

Coefficient of Variation

A disadvantage of the standard deviation as a comparative measure of variation is that it depends on

the units of measurements. This means that it is difficult to use the standard deviation to compare

measurements from different populations. For this reason we have the coefficient of variation which

expresses the standard deviation as a percentage of the sample or population mean.

For Samples For Populations

Example – Big Blossom Greenhouse was commissioned to develop an extra-large rose for the Rose Bowl

Parade. A random sample of blossoms from Hybrid A bushes yielded the following diameters (in inches)

for mature peak blooms. Find the Coefficient of Variation.

2, 3, 3, 8, 10, 10

Solution:

100x

sCV 100

CV


Section 2.4 – Measures of Variation

Objectives

• Determine the range of a data set

• Determine the variance and standard deviation of a population and of a sample

• Use the Empirical Rule and Chebychev’s Theorem to interpret standard deviation

• Approximate the sample standard deviation for grouped data

Three measures of variation:

range

variance

standard deviation

Range

• The difference between the maximum and minimum data entries in the set.

• The data must be quantitative.

• Range = (Max. data entry) – (Min. data entry)

Sample Variance and Sample Standard Deviation

Sample Variance – Can be thought of as a kind of average of the values values. However we

divide by (n-1) instead of n for technical reasons.

Sample Standard Deviation – Can be thought of as a measure of variability or risk. Larger values of s

imply greater variability in the data. Why do we take the square root? Because the units before the

square root are so we take the square root to get back to the original units.

The standard deviation is the square root of the variance.

2s s = standard deviation

s has the same dimensions as the original x’s

How to With your Calculator:

Press STAT ⟶ EDIT

This is where you enter your data

To Analyze your Data:

Press STAT⟶CALC⟶1-Var Stats


Population Variance and Population Standard Deviation

Interpreting Standard Deviation

• Standard deviation is a measure of the typical amount an entry deviates from the mean.

• The more the entries are spread out, the greater the standard deviation.

Population Sample

µ - Population Mean – Sample Mean

σ – Population Standard Deviation s – Sample Standard Deviation

– Population Variance – Sample Variance


Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule)

For data with a (symmetric) bell-shaped distribution, the standard deviation has the following

characteristics:

• About 68% of the data lie within one standard deviation of the mean.

• About 95% of the data lie within two standard deviations of the mean.

About 99.7% of the data lie within three standard deviations of the mean.

Example: Big Blossom Greenhouse was commissioned to develop an extra-large rose for the Rose Bowl

Parade. A random sample of blossoms from Hybrid A bushes yielded the following diameters (in inches)

for mature peak blooms. Find some descriptive statistics for the given data.

2, 3, 3, 8, 10, 10

Solution:

x s 2x s 3x sx3x s x s2x s

68% within 1 standard deviation

34% 34%

99.7% within 3 standard deviations

2.35% 2.35%

95% within 2 standard deviations

13.5% 13.5%

𝜇 3𝜎

Or

𝜇 2𝜎

Or

𝜇 𝜎

Or

𝜇

Or

𝜇 + 𝜎

Or

𝜇 + 2𝜎

Or

𝜇 + 3𝜎

Or


Example: Using the Empirical Rule

In a survey conducted by the National Center for Health Statistics, the sample mean height of women in

the United States (ages 20-29) was 64.3 inches, with a sample standard deviation of 2.62 inches.

Estimate the percent of the women whose heights are between 59.06 inches and 64.3 inches.

Solution:

Critical Thinking

• Standard deviation or variance, along with the mean, gives a better picture of the data

distribution than the mean alone.

• Chebyshev’s theorem works for all kinds of data distribution.

• Data values beyond 2.5 standard deviations from the mean may be considered as outliers.


Example – “Students who care” is a student volunteer program in which college students donate work

time to various community projects. For a random sample of students in the program, the mean number

of hours was 2 hours each semester with a sample standard deviation of hours each

semester. Find an interval to A to B for the number of hours volunteered into which at least 75% of the

students in this program would fall.

Solution: Chevbyshev’s Theorem states that 75 % of the data must fall within 2 standard deviations of

the mean. The mean 2 and the standard deviation the interval is:

2 + 2

2 2 2 + 2

25.7 to 32.5

At least 75% of the students would fall into the group that volunteered from 25.7 to 32.5 hours each

semester.

Standard Deviation for Grouped Data

Sample standard deviation for a frequency distribution

When a frequency distribution has classes, estimate the sample mean and standard deviation by using

the midpoint of each class

2( )

1

x x fs

n

where n= Σf (the number of entries in the data set)


Example: Finding the Standard Deviation for Grouped Data

You collect a random sample of the number of children per

household in a region. Find the sample mean and the sample

standard deviation of the data set.

\

Solution:

Step1: First construct a frequency distribution.

x x2( )x x 2( )x x f

2( ) 145.40x x f

Step 3: Determine the sum of

squares

Step 2: Find the mean of the frequency distribution

The sample mean is about 1.8 children.

Step 4:

Find the sample standard deviation.

𝑠 𝑥−�� 2𝑓

𝑛−1

145 40

50−1 ≈

The standard deviation is about 1.7

children.


Percentiles and Quartiles

• For whole numbers P, 1 ≤ P ≤ 99, the Pth percentile of a distribution is a value such that P% of the

data fall below it, and (100-P)% of the data fall at or above it.

• Q1 = 25th Percentile

• Q2 = 50th Percentile = The Median

• Q3 = 75th Percentile

Fractiles Summary Symbols

Quartiles Divide a data set into 4 equal parts Q1, Q2, Q3

Deciles Divide a data set into 10 equal parts D1, D2, D3, …

Percentiles Divide a data set into 100 equal parts P1, P2, P3, …

Quartiles and Interquartile Range (IQR) : 1

Quantiles divide the data into four parts

To find quantiles

(1.) Arrange the data from smallest to largest and find the median.

(2.) First quantile (Q1) is the median of the observations whose position in the ordered list is to the

left of the median.

(3.) Third quantile (Q3) is the median of the observations whose position in the ordered list is to the

right of the median.

A measure of spread based on Quantiles is the Interquantile range = IQR

IQR =Q3 – Q1

The IQR gives the spread of the middle 50% of the data.


Computing Quartiles

Example – Find the mean ( ) and 1 and the interquartile range (IQR) of the following data.

32 41 59 68 72 78 81 92 95 96

Solution:

Box-and-whisker plot

• Exploratory data analysis tool.

• Highlights important features of a data set.

• Requires (five-number summary):

Minimum entry

First quartile Q1

Median Q2

Third quartile Q3

Maximum entry

Drawing a Box-and-Whisker Plot

1. Find the five-number summary of the data set.

2. Construct a horizontal scale that spans the range of the data.

3. Plot the five numbers above the horizontal scale.

4. Draw a box above the horizontal scale from Q1 to Q3 and draw a vertical line in the box at Q2.

5. Draw whiskers from the box to the minimum and maximum entries.

2( ) 145.40x x f

Whisker Whisker

Maximum entry

Minimum entry

Box

Median, Q2 Q

3 Q

1


Example - Which of the following box-and-whiskers plots suggests a symmetric data distribution?

Percentiles and Other Fractiles

How to compute percentiles given data:

1. Order the data set in ascending order 2. Use this equation to find the index of the percentile: p/100 * (n + 1) where p is the

percentile you want so if the problem asks for P30 then you want the 30th percentile. 3. If the solution is an integer, say 32, then the pth percentile value is the 32nd data point from the

sample in ascending order. If the solution is not an integer then you find the average value be the two surrounding points. For example, say the solution is 23.7 for the index of the pth percentile. The value reported is the average of the 23rd and 24th data point. This can be either the simple average or a weighted average.

a

.

b

.

c

.

d

.


Example: Interpreting Percentiles

The ogive represents the cumulative frequency distribution for SAT test scores of college-bound

students in a recent year. What test score represents the 62nd percentile? How should you interpret

this? (Source: College Board)

Standard Score (z-score)

• Represents the number of standard deviations a given value x falls from the mean μ.

Solution: The 62nd percentile corresponds to a test score

of 1600.

This means that 62% of the students had an SAT score of

1600 or less.

value - mean

standard deviation

xz


If the variable Z (Standard Normal) has a Normal distribution with mean 0 and standard deviation 1 then

we say that

Standardizing Observations and Variables

To find areas under the normal curve we will use the fact that if X is N(µ,σ) then the transformed

variable Z, has a standard normal distribution.

Mean = 0 and Standard deviation 1, this is called the standard normal distribution and −

is called

a standardized variable.

So to compute areas under the curve we will standardize to curve and compute the area

under the curve. So we only need the area under the curve to compute the area under

any Normal curve.

Raw Scores and z Scores

𝑁


Distribution of z-Scores

Example: Comparing z-Scores from Different Data Sets

In 2009, Heath Ledger won the Oscar for Best Supporting Actor at age 29 for his role in the movie The

Dark Knight. Penelope Cruz won the Oscar for Best Supporting Actress at age 34 for her role in Vicky

Cristina Barcelona. The mean age of all Best Supporting Actor winners is 49.5, with a standard deviation

of 13.8. The mean age of all Best Supporting Actress winners is 39.9, with a standard deviation of 14.0.

Find the z-score that corresponds to the ages of Ledger and Cruz. Then compare your results.

Solution:

• If the original x values are normally distributed, so are the z scores of these x values. – µ = 0 – σ = 1

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