Frequency Distributions and Graphs Chapter 2: Frequency Distributions and Graphs

Section 1 – Organizing Data Learning Target - I will be able to organize data using a frequency distribution.

Frequency Distribution The organization of raw data in table form, using classes and frequencies. Class – a qualitative or quantitative category Frequency – the number of data values contained in a specific class

Example: 49 57 38 73 81 74 59 76 65 69 54 56 68 78 85 61 48 37 43 82 64 67 52 77 79 40 80 60 71 83 90 87 Class Limits Tally Frequency 35 – 41 /// 3 42 – 48 49 – 55 //// 4 56 – 62 //// //// 10 63 – 69 70 – 76 5 77 – 83 84 - 90 Total: 50

Categorical Frequency Distribution Used for data that can be placed in specific categories such as nominal and ordinal level data

Example: Raw Data – Blood Types Step 1: Make a Table Steps to Make a Categorical Frequency Distribution A B AB O Step 1: Make a Table Class Tally Frequency Percent A B O AB

Steps continued Step 2: Tally the data Step 3: Count the tallies and place results under frequencies. Step 4: Find the percentage of each value by using the following formula % = f/n x 100% Step 5: Find the totals for the frequency and percent column. Class Tally Frequency Percent A //// 5 20 B //// // 7 28 O //// //// 9 36 AB 4 16 Total: 25 Total: 100

Grouped Frequency Distribution Use when the range of data is large and must be grouped into classes that are more than one unit in width

Example Class Limits Class Boundaries Tally Frequency 24 – 30 23.5 – 30.5 /// 3 31 – 37 30.5 – 37.5 / 1 38 – 44 37.5 – 44.5 //// 5 45 – 51 44.5 – 51.5 //// //// 9 52 – 58 51.5 – 58.5 //// / 6 59 – 65 58.5 – 65.5 Total: 25

Vocabulary for Grouped Frequency Distributions Class Limits Lower class limits – the smallest data value that can be included in the class Upper class limits – the largest data value that can be included in the class Class Boundaries – numbers to separate the classes so there are no gaps Class Width – subtract the lower class limit of the first class from the lower class limit of the second class

Steps to Make a Grouped Frequency Distribution Step 1: Determine the classes Find the highest and lowest values Find the range Select number of classes desired (5 – 20) Find the width by dividing the range by the number of classes and rounding up Select a starting point (usually the lowest value or any convenient number less than the lowest value); add the width to get the lower limits Find the upper class limits Find the boundaries

Steps (cont’) Step 2: Tally the data Step 3: Find the numerical frequencies from the tallies, and find the cumulative frequencies

Reasons for constructing To organize the data in a meaningful, intelligible way. To enable the reader to determine the nature or shape of the distribution To facilitate computational procedures for measures of average and spread 4. To enable the researcher to draw charts and graphs for the presentation of data. 5. To enable the reader to make comparisons among different data sets.

Applying the Concepts 2-1

Histograms, Frequency Polygons, and Ogives Section 2-2 Histograms, Frequency Polygons, and Ogives

Learning Target I will be able to represent data in a frequency distribution graphically using histograms, frequency polygons, and ogives.

Histograms A graph that displays data with contiguous (touching) vertical bars of various heights to represent the frequencies of the classes.

Histograms (cont’) How to create a histogram Step 1: Draw and label the x and y axes. Step 2: Represent the frequency on the y-axis and class boundaries on the x-axis. Give the graph a title. Step 3: Using the frequencies as heights, draw vertical bars for each class.

Example - Histograms Class Boundaries Frequency 99.5 – 104.5 2 104.5 – 109.5 8 109.5 – 114.5 18 114.5 – 119.5 13 119.5 – 124.5 7 124.5 – 129.5 1 129.5 – 134.5

Frequency Polygons A graph that displays the data by using lines that connect the points plotted for the frequencies at the midpoint of the classes

Steps to Create a Frequency Polygon Step 1: Find the midpoints of each class Add the upper and lower boundaries and divide the sum by 2. Step 2: Draw the x and y axes. Label the x with the midpoint and use a suitable scale on the y axis for the frequency. Step 3: Using the midpoints for the x values and the frequencies as y values, plot the points.

Steps (cont’) Step 4: Connect adjacent points with line segments. Draw a line back to the x axis at the beginning and end of the graph, at the same distance that the previous and next midpoints would be located. To do this, add a class to the beginning and end of the frequency distribution with a frequency of zero for both.

Example – Frequency Polygon Class Boundaries Midpoints Frequency 99.5 – 104.5 102 2 104.5 – 109.5 107 8 109.5 – 114.5 112 18 114.5 – 119.5 117 13 119.5 – 124.5 122 7 124.5 – 129.5 127 1 129.5 – 134.5 132 Example – Frequency Polygon

A histogram and frequency polygon are two ways to represent the same data set. The choice of which one to use is up to the researcher.

Ogive – (pronounced ojive) A line graph that represents the cumulative frequencies for the classes in a frequency distribution. Cumulative frequency – the sum of the frequencies accumulated up to the upper boundary of a class in the distribution.

Ogives look like

Steps to Create Ogives Step 1: Find the cumulative frequency for each class. Add each frequency one at a time so that the last class has the total. Step 2: Draw the x and y axes. Label the x axis with the class boundaries. Use an appropriate scale for the y axis to represent the cumulative frequencies.

Steps (cont’) Step 3: Plot the cumulative frequency at each upper class boundary. Step 4: Starting with the first upper class boundary connect adjacent points with line segments. Extend the graph to the first lower class boundary on the x axis.

Example - Ogive Cumulative Frequency Less than 99.5 Less than 104.5 2 Less than 104.5 2 Less than 109.5 10 Less than 114.5 28 Less than 119.5 41 Less than 124.5 48 Less than 129.5 49 Less than 134.5 50

Cumulative frequency graphs are used to visually show how many values are below a certain upper class boundary. To find how many values are less than a specific boundary, draw a vertical line up to the graph and then a horizontal line to the y axis.

Relative Frequency Graphs Histograms, frequency polygons and ogives that use proportions of the raw data. To find the proportion, divide the frequency by the total items. The cumulative relative frequency will always add to one.

Class Boundaries Midpoint Frequency Relative Frequency Cumulative Frequency Cumulative Relative Frequency 5.5 – 10.5 8 1 0.05 Less than 5.5 0.00 10.5 – 15.5 13 2 0.10 Less than 10.5 15.5 – 20.5 18 3 0.15 Less than 15.5 20.5 – 25.5 23 5 0.25 Less than 20.5 6 0.30 25.5 – 30.5 28 4 0.20 Less than 25.5 11 0.55 30.5 – 35.5 33 Less than 30.5 15 0.75 35.5 – 40.5 38 Less than 35.5 0.90 Less than 40.5 20 1.00

Applying Concepts 2-2

Section 2-3 Other Types of Graphs

Bar Graphs Represent the data by using vertical or horizontal bars, whose heights and lengths represent the frequencies of the data

How to create bar graph Draw and label x and y axes. Horizontal – frequency goes on the x axis, categories go on the y axis Vertical – frequency goes on the y axis, categories go on the x axis Draw the bars corresponding to the frequencies.

Example – Horizontal Bar Graph

Example – Vertical Bar Graph

Pareto Charts Represent a frequency distribution for a categorical variable, and the frequencies are displayed by the heights of vertical bars, which are arranged in order from highest to lowest

How to create a Pareto Chart Arrange the data from highest to lowest according to frequency. Draw and label the x and y axes. The x axis is the category and the y axis is the frequency. Draw the bars corresponding to the frequencies. Make sure the bars are touching.

Example – Pareto Chart

Time Series Graph Represents data that occur over a specific period of time

How to create a Time Series Graph Draw and label the x and y axes. The x axis is the time and the y axis is the data. Plot each point from the table. Draw a line connecting the points. The line does not have to be a smooth curve or straight line, just connect the dots.

Example – Time Series Graph

Pie Graph A circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.

How to create a Pie Graph Since there are 360 degrees in a circle, the frequency for each class must be converted into a proportional part of the circle. This is done by using the formula 𝑑𝑒𝑔𝑟𝑒𝑒𝑠= 𝑓 𝑛 × 360 where f is the frequency and n is the sum of the frequencies. Convert each frequency to a percentage by using the formula %= 𝑓 𝑛 ×100%

Steps continued 3. Use a protractor and a compass to draw the graph using the appropriate degree measures and label the sections with the name and percentages of each category

Example – Pie Chart

Stem and Leaf Plots A data plot that uses part of the data value as the stem and part of the data value as the leaf to form groups or classes.

How to create a Stem and Leaf Plot Arrange the data in order from lowest to highest. Separate the data according to the first digit. In a table with two columns, the left column being the stem and the right being the leaves, place the first digit of the groups in the stem column and the digit that follow in the leaf column.

Things to remember Only one number can be a leaf Stems can be multiple digits Leaves are arranged in order from lowest to highest, separated by a space or comma

Back to Back Stem and Leaf Plot If you are comparing two sets of data, you can use a back to back stem and leaf plot A back to back stem and leaf plot has three columns. The middle column is the stem, the left column is the leaves of the first set of data with the leaves being arranged from right to left, lowest to highest. The right column is the leaves of the second set of data with the leaves arranged from left to right, lowest to highest.

Example – Stem and Leaf Plot Leaves 2 1 3 4 0 3 5 3 1 2 2 2 2 3 6 4 3 4 4 5 5 1 2 7

Example – Back to Back Stem and Leaf Plot Atlanta Stem Philadelphia 9 8 6 2 5 8 6 4 4 2 2 2 2 2 1 3 0 0 0 0 2 2 3 4 6 6 6 8 8 9 9 7 4 4 0 0 4 0 0 0 0 5 3 2 2 0 0 0 3 4 8 6 1 7

Applying the Concepts 2-3

Answers – Applying the Concepts 2-3 year, cause of death, and rate of death per 100,000 men Cause of death is qualitative, year and death rates are quantitative Year is discrete, death rate is continuous, cause of death is neither Line graph

Answers continued 5. No, pareto charts can only have one qualitative and one quantitative variable 6. No, same reason as pareto chart 7. Pareto chart typically shows categorical variable listed from highest frequency category to the lowest frequecy category 8. Time series chart is used to see trends in the data, can also be used for forecasting and predicting.