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Probability and Statistics Chapter 2: Descriptive Statistics

PART 1 – Organizing Data

Pd 2 Pd 4 Pd 7 In Class Homework

Mon 8/8

Mon 8/8

Mon 8/8

2.1A – Frequency Distributions and Their Graphs [Histograms]

2.1A HW pg 49-51 #16,18,19,20,29 [5Q’s]

Tue 8/9 A

Wed 8/10 B

Wed 8/10 B

2.1B – Frequency Distributions and Their Graphs [Histograms, Frequency Polygon,Ogive]

2.1B HW Pg 50-52 #22,24,26,32,36 [6Q’s]

Wed 8/10 B

Thur 8/11 C

Thur 8/11 C

HW 2.1A and 2.1B DUE TODAY 2.2A – More Graphs and Displays [Stem and Leaf, Dotplot, Time Series]

2.2A HW Pg 63-66 #17,20,21,23,24,31,41 [7Q’s]

Fri 8/12

Fri 8/12

Fri 8/12

2.2B – More Graphs and Displays [Bar, Pareto, Circle]

2.2B HW Pg 64 #25,27 and pg 120 #3 [4Q’s—because #3 has 2 graphs]

Mon 8/15

Mon 8/15

Mon 8/15

HW 2.2A and 2.2B DUE TODAY 2.3A – Measures of Central Tendency [Distribution Shapes]

2.3A HW INSTRUCTIONS: Do not follow textbook instructions. Instead, for each graph, describe the shape of the distribution Pg 74-77 #9-12,31-40 [14Q’s]

Tue 8/16 A

Wed 8/17 B

Wed 8/17 B

HW 2.3 A DUE TODAY Review Chapter 2 – Part 1

Test Review Study Guide Worksheet

Wed 8/17 B

Thurs 8/18 C

Thurs 8/18 C

TEST REVIEW STUDY GUIDE DUE TODAY TEST CHAPTER 2 – PART 1 Covers only sections 2.1, 2.2, 2.3A

Goal in this unit: 1. Examine data & describe the distribution of the data 2. Choose the best way to organize/display the data 3. Create (by hand and using tech) the most common data displays 4. Read/Interpret data displays

Academic Standards:

PS.DA.1 Create, compare, and evaluate different graphic displays of the same data, using histograms, frequency polygons, cumulative frequency

distribution functions, pie charts, scatterplots, stem-and-leaf plots, and box-and-whisker plots. Draw these with and without technology.

PS.DA.2 Compute and use mean, median, mode, weighted mean, geometric mean, harmonic mean, range, quartiles, variance and standard

deviation. Use tables and technology to estimate areas under the normal curve. Fit a data set to a normal distribution and estimate

population percentages. Recognize that there are data sets not normally distributed for which such procedures are inappropriate.

PS.DA.5 Recognize how linear transformations of univariate data affect shape, center, and spread.

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2.1 Frequency Distributions and Their Graphs Have a large set of quantitative data? Organize into smaller intervals called ________________.

FREQUENCY DISTRIBUTION A frequency distribution is a table that shows a count of the number of entries in each interval or count.

Each class has a LOWER CLASS LIMIT: the __________________ number that can belong to the class. Each class has an UPPER CLASS LIMIT: the __________________ number that can belong to the class.

The CLASS WIDTH is the distance between lower limits of consecutive classes.

The class width of this frequency distribution is: ______

EXPANDED FREQUENCY DISTRIBUTION An expanded frequency distribution table shows even more data.

CLASS BOUNDARIES: numbers that separate classes without forming gaps. Lower Class Boundary = Lower Class Limit – 0.5 Upper Class Boundary = Upper Class Limit + 0.5

MIDPOINT: the sum of the lower and upper limits of the class divided by two

RELATIVE FREQUENCY: 𝐶𝑙𝑎𝑠𝑠 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦

𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒=

𝑓

𝑛

CUMULATIVE FREQUENCY: The sum of the frequency of that class and all previous classes. The cumulative frequency of the last class = the sample size, n.

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TITLE

Freq

/Rel

Fre

q./

Etc.

Class Boundaries

HISTOGRAM A histogram uses ______________ to show the frequency of classes. A relative frequency histogram uses bars to show the _____________________ of cases in each class. Basic Construction: Characteristics of a Histogram: 1. Used for high volume quantitative data 2. Bars equal width 3. Bars touch 4. Class boundaries on x-axis 5. Class frequency/relative frequency on y-axis 6. Classes cannot overlap or be open-ended

7. Use 5 to 20 classes Example 1:

This histogram has _____ CLASSES.

The CLASS BOUNDARIES of this bar are _______ to ______.

The FREQUENCY of this class is ___________. OK. Sounds good. Now how do we make a frequency distribution table and a histogram? I knew you were going to ask that. Here we go…

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Example 1: Use the given frequency distribution to construct an expanded frequency distribution showing Class Boundaries, Midpoints, Relative Frequency, and Cumulative Frequency. Getting Started: What is the class width?

Example 2: Construct a Frequency Distribution for the given number of classes. Include the class boundaries, the midpoints, the relative frequencies, and the cumulative frequencies. An irate customer called the Dollar Day Mail Order Company 40 times during the last two weeks to see why his order had not arrived. Each time he called, he recorded the length of time he was put “on hold” before begin allowed to talk to a customer service representative.

Number of Classes: 5

Step 1: Determine Class Width In our example:

Class width = 𝑅𝑎𝑛𝑔𝑒 𝑜𝑓 𝐷𝑎𝑡𝑎

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑙𝑎𝑠𝑠𝑒𝑠

ROUND UP to next convenient number.

*RANGE = largest value – smallest value

Time on Hold, in minutes

1 5 5 6 7 4 8 7 6 5

5 6 7 6 6 5 8 9 9 10

7 8 11 2 4 6 5 12 13 6

3 7 8 8 9 9 10 8 9 9

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Table repeated for convenience. Class width: _____

Step 2: Determine the Class Limits

Lowest data value = Lower class limit of the first class. Add Class Width to get next lowest limit, etc. Use logic to determine the upper limit of the first class. Add Class Width to get the next upper limit, etc.

Step 3: Determine the Class Boundaries Step 6: Calculate Relative Frequency Step 4: Determine the Midpoint of each class. Step 7: Calculate the Cumulative Frequency. Step 5: Determine the Frequency of each class. (tally chart?)

CONSTRUCTING A HISTOGRAM Horizontal Axis: Class Boundaries Vertical Axis: Frequency Axes must have SCALE and labels. Graph should have a title. Example 3:

Time on Hold, in minutes

1 5 5 6 7 4 8 7 6 5

5 6 7 6 6 5 8 9 9 10

7 8 11 2 4 6 5 12 13 6

3 7 8 8 9 9 10 8 9 9

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Example 4: Construct a frequency distribution table and a histogram using six classes for the following data. Retirement Ages of 24 doctors: 70 54 55 71 57 58 63 65

60 66 57 62 63 60 63 60

66 60 67 69 69 52 61 73

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RELATIVE FREQUENCY HISTOGRAM

A relative frequency histogram uses the PERCENTAGE 𝑓

𝑛 for

each class on the vertical axis instead of the frequency of each class. Example 5: Notice instead of class boundaries, the midpoints have been used on the horizontal axis. What is the class width? What are the class boundaries of the first class? Which class has the lowest relative frequency? Example 6: Construct a relative frequency histogram

Vertical Axis: Relative Frequency Horizontal Axis: Class boundaries *Don’t forget! Every graph should have a title and have axes labeled and properly scaled.*

Relative Frequency

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FREQUENCY POLYGON Sometimes a frequency polygon is used to describe a frequency distribution. It is used to emphasize the

continuous change in frequency. Vertical Axis: Frequency Horizontal Axis: Class Midpoint (Extend 1 class width to left and right) Plot points: (Midpoint, Frequency) Example 7: a) What is the class width?

b) Which class had the greatest frequency? c) Which class had the least frequency? Example 8:

What is the class width? Leftmost Scale Value: Rightmost Scale Value:

Midpoint

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OGIVE (Cumulative Frequency Graph) pronounced: oh-jive An ogive is a line graph. Vertical Axis: cumulative frequency Horizontal: Upper class boundary (extend left 1 class width) Points: (UB, CF)

Example 9: Consider the Ogive above. a) What is the sample size?

b) What is the cumulative frequency for a GPS navigator that costs $170.50? c) What is the price for with the cumulative frequency is approximately 19? d) Estimate the number of GPS navigators between $282.50 and $338.50. e) Estimate the number of GPS navigators greater than $226.50.

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2.2 More Graphs and Data Displays A histogram is a great way to organize and display large quantities of quantitative data. If you have a small or medium amount of data, you may want to use a Stem & Leaf Diagram or a Dot Plot.

Stem and Leaf Plot

Each number is separated into a STEM and a LEAF o Leaves: One Digit, Numerical Order o Stems: Basically everything but the last digit; Include all stems from lowest to highest

even if a particular stem does not have any leaves. o YOU MUST INCLUDE A KEY!

Similar to a Histogram

Still contains original values

Data Values: Data Values: Data Values: Example 1: Construct a Stem-and-Leaf Plot The numbers of home runs that Sammy Sosa hit in the first 15 years of his major league baseball career are listed below. Make a stem-and-leaf plot for this data. What can you conclude about the data?

4 15 10 8 33

25 36 40 36 66

63 50 64 49 40

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Example 2: The GPAs of 20 students are listed. Make a stem-and-leaf plot for this data.

Sometimes you may have too many leaves on a stem to really see the pattern of the data. You can use TWO ROWS for each stem. The first row will be for leaves 0 to 4, the second row for leaves 5 to 9. Example 3: Use a stem-and-leaf plot that has two rows for each stem to display the data.

108 104 105 109 102

112 114 118 100 121

125 130 124 118 111

113 106 111 129 122

122 107 102 132 128

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Dot Plots Another type of display for quantitative data is a dot plot. Each value is plotted as a dot above a scaled horizontal axis.

A dot plot can be created like this too: A dot plot can be used to tell a story, much like a

histogram

Example 4: Use a dot plot to display the data.

Horizontal Axis Only

Should have SCALE

Dots should be equally spaced vertically.

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Histograms, Stem-and-Leaf Plots, and Dot Plots all deal with UNIVARIATE or _____________________ data. Sometimes we have to deal with BIVARIATE DATA. Bivariate data occurs when each value in a data set is __________________ with another entry in a second data set. For example, in Algebra 2, you learn about bivariate data when you made scatter plots.

TIME SERIES GRAPHS A time-series graph is used for displaying quantitative measurements taken over regular intervals of time. This is a type of bivariate data. A time-series graph is a type of line graph. Example 5: The table lists the number of cellular telephone subscribers (in millions) for the years 1998 through 2007. Construct a time series chart for the number of cellular subscribers. (Source: Cellular Telecommunication & Internet Association)

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BAR GRAPH

Clustered Bar Graph: two or more bars for each value on the horizontal axis, clusters are uniformly spaced

PARETO CHART A type of bar graph in which the bars are arranged in decreasing order. Very useful for Qualitative Data!

Example 6: A study is conducted to see how people get jobs. Four hundred people were randomly selected and interviewed. Create a Pareto Chart to display this data.

Features of a Bar Graph

Bars can be vertical or horizontal.

Bars are of uniform width and uniformly spaced.

Lengths represent frequency or relative frequency of occurence. The same measurement scale is used for the length of each bar.

The graph is well annotated with title, labels for each bar, and vertical scale or actual value for the length of each bar.

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CIRCLE GRAPHS/PIE CHARTS Wedges visually display proportional parts of the total population as a percentage or as a portion of 360° Good for qualitative/categorical data The graph should have a title and wedges should be well labeled or have a key/legend. Josh Sundquist’s Pie Charts for Math Nerds: http://youtu.be/LhfGPqW2xkM How do you make a circle graph by hand? Step 1: Determine your sample size. (“n”= the grand total) Step 2: Determine the relative frequency of each category ( rf = f÷n ) Step 3: Determine the number of DEGREES represented by each category Degrees of category = rf x 360° Step 4: Use a PROTRACTOR to mark off the correct number of degrees, one wedge at a time Example 7: A study was conducted to determine how certain families pay on their credit card balances. Two hundred families were randomly selected and the results are listed below. Construct a pie chart of the data.

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Example 8: Make a circle graph for the following data

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2.3A – Measures of Central Tendency – Distribution Shapes Recall a __________________________ can be used to represent a Frequency Distribution. Symmetric/Mound/Bell Shaped: two sides are symmetrical with respect to a vertical line that goes through the middle of the graph

Uniform: every class has the same frequency Bimodal: histogram shows _______ peaks separated by at least one shorter bar Unimodal: histogram shows ______ peak Skewed Left: More bars on the left side of the peak… “tail” on the left is longer than right Skewed Right: More bars on the right side of the peak….“tail” on right is longer than left Often a ______________________ distribution is caused by collecting data from a group of individuals that could have been classified better into two separate groups for that particular data. Example: height from a mixed group of men and woman Significant gaps between bars at the left or right can be caused by _______________________. These are values that are significantly higher or lower than the rest of your data. Example: salaries of employees at a major corporation where the CEO makes three times as much as rest of the workers.

2 | 14Copyright © Cengage Learning. All rights reserved.

Distribution Shapes

Symmetric Uniform

Skewed Left Skewed Right

Bimodal

2 | 14Copyright © Cengage Learning. All rights reserved.

Distribution Shapes

Symmetric Uniform

Skewed Left Skewed Right

Bimodal

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Example 1: NAME THAT DISTRIBUTION! Unimodal, Bimodal, Uniform?

Symmetrical, Skewed Left, Skewed Right? Possible Outlier?

a) b) c) d)

e) f)

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g) h) i) j) ***** PLEASE NOTE CHANGE IN DIRECTIONS FOR DISTRIBUTION SHAPES HW 2.3A******* SEE ASSIGNMENT GUIDE FOR SPECIFICS!!!