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Descriptive Statistics

What a Data Set Tells Us

15.1 Organizing and Visualizing DataObjectives:

- Understand the difference between a sample and a population.
- Organize data in a frequency table.
- Use a variety of methods to represent data visually.
- Use stem-and-leaf displays to compare data.

Populations and Samples

- Statistics is an area of mathematics in which we are interested in gathering, organizing, analyzing, and making predictions from numerical information called data.
- The set of all items under consideration is called the population. Often only a sample or subset of the population is considered.

Sample bias

- We will describe a sample as biasedif it does not accurately reflect the population as a whole with regard to the data that we are gathering.
- Bias could occur because of the way in which we decide how to choose the people to participate in the survey. This is called selection bias.
- Another issue that can affect the reliability of a survey is the way we ask the questions, which is called leading-question bias.

Frequency Tables

We show the percent of the time that each item occurs in a frequency distribution using a relative frequency distribution.

We often present a frequency distribution as a frequency table where we list the values in one column and the frequencies of the values in another column.

Frequency Tables cont. (2)

Example: 25 viewers evaluated the latest episode of CSI. The possible evaluations are

(E)xcellent, (A)bove average, a(V)erage, (B)elow average, (P)oor.

After the show, the 25 evaluations were as follows:

A, V, V, B, P, E, A, E, V, V, A, E, P, B, V, V, A, A, A, E, B, V, A, B, V

Construct a frequency table and a relative frequency table for this list of evaluations.

Frequency Tables cont. (3)

Solution: We organize the data in the frequency table shown below.

We construct a relative frequency distribution for these data by dividing each frequency in the table by 25. For example, the relative frequency

of E is .

Frequency Tables cont. (4)

Example: Suppose 40 health care workers take an AIDS awareness test and earn the following scores:

79, 62, 87, 84, 53, 76, 67, 73, 82, 68,

82, 79, 61, 51, 66, 77, 78, 66, 86, 70,

76, 64, 87, 82, 61, 59, 77, 88, 80, 58,

56, 64, 83, 71, 74, 79, 67, 79, 84, 68

Construct a frequency table and a relative frequency table for these data.

Frequency Tables cont. (5)

Solution: Because there are so many different scores in this list, the data is grouped in ranges.

To get the relative frequency, we divide each count in the frequency column by 40. For example, in the row labeled 55–59, we divide 3 by 40 to get 0.075 in the third column.

Representing Data Visually

- A bar graphis one way to visualize a frequency distribution. In drawing a bar graph, we specify the classes on the horizontal axis and the frequencies on the vertical axis.
- If we are graphing a relative frequency distribution, then the heights of the bars correspond to the size of the relative frequencies. Graphing the relative frequencies, rather than the actual values in two data sets, allows us to compare the two distributions.

Representing Data Visually cont. (2)

Example: Draw a bar graph of the frequency distribution of viewers’ responses to an episode of CSI.

Solution:

Representing Data Visually cont. (3)

Example: Draw a bar graph of the relative frequency distribution of viewers’ responses to an episode of CSI.

Solution:

Discrete or Continuous Variables

- A variable quantity that cannot take on arbitrary values is called discrete. Other quantities, called continuousvariables, can take on arbitrary values.
- The number of children in a family is an example of a discrete variable. Weight is an example of a continuous variable.
- We use a special type of bar graph called a histogramto graph a frequency distribution when we are dealing with a continuous variable quantity or a variable quantity that is discrete, but has a very large number of different possible values.

Histograms

Example: A clinic has the following data regarding the weight lost by its clients over the past 6 months. Draw a histogram for the relative frequency distribution for these data.

Solution: We first find the relative frequency distribution.

Histograms cont. (2)

We draw the histogram exactly like a bar graph except that we do not allow spaces between the bars.

Bar graphs

Example: The bar graph shows the number of Atlantic hurricanes over a period of years.

What was the smallest number of hurricanes in a year during this period? ___

b) What was the largest? ___

What number of hurricanes per year occurred most frequently? ___

How many years were the hurricanes counted? ___

In what percentage of the years were there more than 10 hurricanes? ___

Stem and Leaf Display

In constructing a stem-and- leaf display, we view each number as having two parts. The left digit is considered the stem and the right digit the leaf.

For example, 38 has a stem of 3 and a leaf of 8.

Example: The following are the number of home runs hit by the home run champions in the National League for the years 1975 to 1989 and for 1993 to 2007.

- 1975–1989: 38, 38, 52, 40, 48, 48, 31, 37, 40, 36, 37, 37, 49, 39, 47
- 1993–2007: 46, 43, 40, 47, 49, 70, 65, 50, 73, 49, 47, 48, 51, 58, 50
Compare these home run records using a stem-and-leaf display.

Stem and Leaf Display cont. (2)

- Solution: Displays should have a title and a key.

1975 to 1989 1993 to 2007

5 | 2 represents 52 home runs 7 | 3 represents 73 home runs

Back-to-back Stem-and-leaf Display

We can compare these data sets by placing these two displays side by side as shown below. Some call this display a back-to-back stem-and-leaf display.

This comparison makes it clear that the home run champions hit significantly more home runs from 1993 to 2007 than from 1975 to 1989.

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