Descriptive Statistics, Part Two

Farrokh Alemi, Ph.D.

Kashif Haqqi, M.D.

Table of Content

• Objectives• Frequency distribution

– Categorical

– Ungrouped

– Grouped

• Histogram• Cumulative frequency• Pie chart• Bar chart

Objectives

1. Given a set of nominal or ordinal data, interpret its frequency distribution, pie chart, and bar chart.

2. Given a set of numerical data, interpret its frequency distribution, cumulative frequency distribution, and histogram.

3. Given a set of numerical data, interpret the meaning of its range, variance, standard deviation, and coefficient of variation.

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Organizing Data

• Data when collected in original form is called “raw data”.

• For example:

Frequency Distribution

• The researches organizes the raw data by using frequency distribution.

• The frequency is the number of values in a specific class of data.

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

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Frequency Distribution Cont.

• For the first data set, a frequency distribution is shown as follow:

Class limits Tally Frequency

1-3 ///// ///// 10

4-6 ///// ///// //// 14

7-9 ///// ///// 10

10-12 ///// / 6

13-15 ///// 5

16-18 ///// 5

Types of Frequency Distribution

• There are three basic types of frequency distribution:– Categorical– Ungrouped – Grouped

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Categorical Frequency Distribution

• The categorical frequency distribution is used for data that can be placed in specific categories, such as nominal or ordinal data.

• For example, data such as political affiliations, religion affiliations, or major field of study would use categorical frequency distribution.

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Example

• The blood type of different students:

Example Cont.

Class Tally Frequency

A ///// 5

B ///// // 7

O ///// //// 9

AB ///// 4

Total 25

Ungrouped Frequency Distribution

• When the range of data is small, the data must be grouped into classes that are not more than one unit in width.

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Example

• The range in the example is R=highest value – lowest value : 11 – 4 = 7

• Since the range is small classes consisting of single data value can be used.

Example Cont.

Class Tally Frequency

4 // 2

5 /// 3

6 / 1

7 ///// 5

8 ///// // 7

9 //// 4

10 // 2

11 / 1

Grouped Frequency Distribution

• When the range of the data is large, the data must be grouped into classes that are more than one unit in width.

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Example

Example Cont.Class limits Tally Frequency

1-3 ///// ///// 10

4-6 ///// ///// //// 14

7-9 ///// ///// 10

10-12 //// / 6

13-15 //// 5

16-18 //// 5

• In this distribution, the values 1 and 3 of the first class are called “class limits”.• 1 is the “lower class limit” and 3 is the “upper class limit.”

Histogram

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

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Histogram in Excel (Steps)

• Open the tools menu, choose the data analysis command and chose histogram from the analysis tools list box.

• Input range: enter the reference for the range of cells containing the data.

• Bin range: enter the reference of cells containing the values that separate the intervals, including the labels. (They must be in ascending order).

Steps Continued

• Labels: check this box to indicate that labels have been included in the references for the input range and bin range.

• Out put range: enter the reference for the cell you want out put to appear.

• Chart output: check this option to obtain a histogram chart in addition to the frequency distribution table on the worksheet.

Create a histogram for the following data:

5, 6, 4, 7, 5, 9, 11, 12, 4, 5, 6, 7, 9, 19

Do this in Excel

Cumulative Frequency

• The cumulative frequency is the sum of the frequencies accumulated up to the upper boundary of a class in the distribution.

• They are used to visually represent how many values are below a certain upper class boundary.

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Example of Cumulative Frequency Distribution

Class FrequencyCumulative frequency

1-4 6 6 5-8 2 8

9-12 5 13 12-16 3 16

Pie Chart

• A pie chart is a circle that is divided into sections according to the percentage of frequencies in each category of the distribution.

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Example

Example Cont.

5%3%8%

51%

33%Convertibles

Station wagons

Compacts

Coupes

Sedans

Bar Chart

• A bar chart is a broader concept than histogram.

• A bar chart may be used to display concepts other than frequency of an observations. For example, a bar chart may display the average exam results.

• Histogram is a bar chart of frequency distribution.

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