CHAPTER 2 Frequency Distributions and Graphs

2-1Introduction 2-2Organizing Data 2-3Histograms, Frequency Polygons, and Ogives 2-4Other Types of Graphs Outline

Organize data using frequency distributions. Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives. Represent data using Pareto charts, time series graphs, and pie graphs. Objectives

2-1 Introduction In a statistical study, data organization and presentation are very important for better understanding and interpretation.

2-2 Organizing Data When data are collected in original form, they are called raw data. When the raw data is organized into a frequency distribution, the frequency will be the number of values in a specific class of the distribution.

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

2-2 Types of Frequency Distribution There are three major type of frequency distribution: i) Categorical Frequency Distribution ii) Ungrouped Frequency Distribution iii) Grouped frequency Distribution

Categorical Frequency Distribution Categorical frequency distribution is used for data that can be categorized into specific group according to its attributes. Example: Gender, blood type, major field of study, religious affiliation, etc.

ClassFrequency A5 B7 AB9 O4 Total25 Example of Categorical Frequency Distribution Blood Type

Ungrouped Frequency Distribution Ungrouped frequency distribution is used for data that can be counted or measured, however the data range is not wide.

ClassFrequency Total26 Example of Ungrouped Frequency Distribution Number of children in a family

Grouped Frequency Distribution Grouped frequency distribution is used for data that can be counted or measured. The range of values in the data set is very large. The data must be grouped into classes that are more than one unit in width. Example: Traveling time in mins, body length in cm, life of boat batteries in hrs, etc.

Example of Grouped Frequency Distribution Lifetimes of Boat Batteries Class limitsClass boundaries FrequencyCumulative frequency Data are classified in a range of values

2-2 Associated Terms for Grouped Frequency Distribution Class Limits Class Boundaries Class Width

Class Limits  Class limits represent the smallest and largest data values that can be included in a class.  Upper class limits The largest value in a class  Lower class limits The smallest value in a class

Class Limits Can you identify the upper and lower class limits for all the classes shown above?

Class Boundaries  Class boundaries used to separate the classes so that there are no gaps in the frequency distribution.

How to find class boundaries? Lower class boundary = (Lower class limit of a class + Upper class limit of the previous class)/2 Upper class boundary = (Upper class limit of a class + Lower class limit of the subsequent class)/2

Class Limits Class 1: Class 2: Class 3: What is the lower class boundary for class 2? What is the upper class boundary for class 2?

Class Width  Class width represents the size of a class.  Class width can be calculated by subtracting the upper class boundary from lower class boundary of the same class.

Lower class limit Upper class limit Class Lower class boundary Upper class boundary Class width

2-2 Guidelines for Constructing a Frequency Distribution 1. The classes must be mutually exclusive Age Mutually exclusive - No overlapping in class limits so that data can only be placed in one class.

2.The classes must be continuous. Age Continuous - There is no gap between different classes.

3. There should be between 5 and 20 classes. 4. The classes must be equal in width.

2-2 Procedures for Constructing a Grouped Frequency Distribution  Find the highest and lowest value.  Find the range (Highest - Lowest).  Select the number of classes desired.  Find the width by dividing the range by the number of classes and rounding up. Step 1

 Select a starting point (usually the lowest value); add the width to get the lower limits.  Find the upper class limits.  Find the boundaries.

 Tally the data. Step 2  Find the frequency. Step 3  Find the cumulative frequency. Step 4

2-2 Example for Grouped Frequency Distribution A book store recorded the number of books sold on 20 consecutive Fridays. Construct a grouped frequency distribution for these

STEP 1: Find the highest and lowest values H = 22 and L = 5 STEP 2: Find the range Range = 22 – 5 = 17 STEP 3: Select the number of classes desired. In this case it is equal to 6.

STEP 4: Find the class width by dividing the range by the number of classes. Width = 17/6 = This value is rounded up to 3. STEP 5: Select a starting point for the lowest class limit. In this case, 5 was chosen as it is the smallest data value. The lower class limits will be 5, 8, 11, 14, 17, and 20.

STEP 6: The upper class limits will be 7, 10, 13, 16, 19, and 22. For example, the upper limit for the first class is 7. STEP 7: Find the class boundaries by subtracting 0.5 from each lower class limit and adding 0.5 to the upper class limit. STEP 8 : Tally the data, write the numerical values for the tallies in the frequency column, and find the cumulative frequencies.

Class LimitsClass Boundaries FrequencyCumulative Frequency 5 – 74.5 – – – – – – – – – –

2-3 Histograms, Frequency Polygons, and Ogives Three most commonly used graphs in research are:  Histogram.  Frequency polygon.  Cumulative frequency graph, or ogive.

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

Example of Histogram Frequency Class boundaries

Histogram VS Bar Chart No gap between the bars There is a gap between the bars

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

Example of Frequency Polygon Frequency Polygon

How to plot a frequency polygon? 1.Plot a histogram. 2.Mark on the mid-point of each class interval. 3.Join all the mid-points with a straight line. Mid-point of an interval = ½ (lower CB + upper CB)

Cumulative Graph / Ogive Cumulative graph or ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution.

Example of Frequency Polygon Ogive

How to plot an ogive? 1.Cumulative frequencies are plotted against the upper class boundaries. 2.The points are joined with a smooth curve.

Question 10 students were asked to solve a simple problem and the time taken by each was noted. Construct a histogram, frequency polygon, and ogive for this data. Time (seconds)Frequency

2-3 Other Types of Graphs Pareto chart is used to 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.

Example of Pareto Chart Bars are arranged from highest to lowest frequency

Time series graph represents data that occur over a specific period of time.

Example of Time Series Chart Changes in number of customers over a certain period of time

Pie chart is a circle divided into sections according to the percentage of frequencies in each category of the distribution. A pie chart should not consist of too many segments (less than eight is suggested).

Example of Pie Chart