BUS 220:ELEMENTARY STATISTICS

Chapter 2: Describing Data

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Summary of Types of Variables

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Summary of the Characteristics for Levels of Measurement

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GOALS• Given a qualitative data set, create by hand

a frequency table, bar chart, and pie chart.

• Given a quantitative data set, be able to prepare by hand frequency distribution, relative frequency distribution, cumulative percentage distribution, histogram, frequency polygon, or a cumulative frequency polygon.

• Use EXCEL to do the above.

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Frequency Table

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Relative Class Frequencies

Class frequencies can be converted to relative class frequencies to show the fraction of the total number of observations in each class.

A relative frequency captures the relationship between a class total and the total number of observations.

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Bar Charts—Column Chart In Excel

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Pie Charts

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Pie Chart Using Excel

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Examples of the use of chart

HMSA Rate Incrase Population of primary voters in New Hampshire

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Examples of the use of chart… Months of remaining inventory of real estate on Oahu

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Frequency Distribution

A Frequency distribution is a grouping of data into mutually exclusive categories showing the number of observations in each class.

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EXAMPLE – Constructing Frequency Distributions: Quantitative Data

Ms. Kathryn Ball of AutoUSA wants to develop tables, charts, and graphs to show the typical selling price on various dealer lots. The table on the right reports only the price of the 80 vehicles sold last month at Whitner Autoplex.

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Constructing a Frequency Table - Example

Step 1: Decide on the number of classes. A useful recipe to determine the number of classes (k) is the “2 to the k rule.” such that 2k > n.There were 80 vehicles sold. So n = 80. If we try k = 6, which means we would use 6 classes, then 26 = 64, somewhat less than 80. Hence, 6 is not enough classes. If we let k = 7, then 27 128, which is greater than 80. So the recommended number of classes is 7.

Step 2: Determine the class interval or width. The formula is: i (H-L)/k where i is the class interval, H is the highest observed value, L is the lowest observed value, and k is the number of classes.($35,925 - $15,546)/7 = $2,911Round up to some convenient number, such as a multiple of

10 or 100. Use a class width of $3,000

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Step 3: Set the individual class limits

Constructing a Frequency Table - Example

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Step 4: Tally the vehicle selling prices into the classes.

Step 5: Count the number of items in each class.

Constructing a Frequency Table

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Excel formula to help us count =COUNTIFS

($A$2:$A$81,">="&G15, $A$2:$A$81,"<"&H15)

Data Range

Lower Limit Condition

Upper Limit Condition

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Class Intervals and Midpoints

Class midpoint: A point that divides a class into two equal parts. This is the average of the upper and lower class limits.

Class frequency: The number of observations in each class.

Class interval: The class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class.

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Class Intervals and Midpoints Example

Referring to the AutoUSA example

Class midpoint: For the first class the lower class limit is $15,000 and the next limit is $18,000. The class midpoint is $16,500, found by:

($15,000 + $18,000)/2

Class interval: The class interval of the vehicle selling price data is $3,000. It is found by subtracting the lower limit of the first class, $15,000, from the lower limit of the next class:

($18,000 - $15,000)

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Relative Frequency Distribution

To convert a frequency distribution to a relative frequency distribution, each of the class frequencies is divided by the total number of observations.

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Graphic Presentation of a Frequency Distribution

The three commonly used graphic forms are:

Histograms Frequency polygons Cumulative frequency distributions

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Histogram

Histogram • A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. • The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other.

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Histogram Using Excel

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Frequency Polygon A frequency polygon and relative

frequency polygon also show the shape of a distribution and is similar to a histogram.

It consists of line segments connecting the points formed by the intersections of the class midpoints and the class frequencies.

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Example

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Frequency Polygon Vs. Relative Frequency Polygon

Both provide quick picture of the main characteristics of the data

Relative frequency polygon is good for comparison between two data set that have quite large difference of total frequency.

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Cumulative Frequency Distribution

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Cumulative Frequency Polygon

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A word about interpreting chart Lowest, Highest number, the largest

concentration Look for trend or cluster of data Based on your experience or past data draw

conclusion or suggestion from the chart.

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Which chart is right?

You want to present the shortage of registered nurses for the state of Hawaii.

Will you use the table?...

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Which chart is right?

Or time-series chart?...

Shortage Projected to Get Worse

0

2,000

4,000

6,000

8,000

10,000

12,000

2005 2010 2015 2020

EstimatedRegistered Nurses

Estimated Need

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Which chart is right?

Or a bar chart…

Shortage Projected to Get Worse

0

2,000

4,000

6,000

8,000

10,000

12,000

2005 2010 2015 2020

Estimated RegisteredNurses

Estimated Need