Section 2-3 Histograms

Learning Objectives We use a visual tool called a histogram to analyze the shape of the distribution of the data.

Histogram A graph consisting of bars of equal width drawn adjacent to each other (without gaps). The horizontal scale represents the classes of quantitative data values The vertical scale represents the frequencies. The heights of the bars correspond to the frequency values.

Histogram Basically a graphic version of a frequency distribution.

Histogram The bars on the horizontal scale are labeled with one of the following: Class boundaries Class midpoints Lower class limits (introduces a small error) Horizontal Scale for Histogram: Use class boundaries or class midpoints. Vertical Scale for Histogram: Use the class frequencies.

Relative Frequency Histogram Has the same shape and horizontal scale as a histogram, but the vertical scale is marked with relative frequencies instead of actual frequencies.

Critical Thinking Interpreting Histograms Objective is not simply to construct a histogram, but rather to understand something about the data. When graphed, a normal distribution has a “bell” shape. Characteristic of the bell shape are (1) The frequencies increase to a maximum, and then decrease, and (2) symmetry, with the left half of the graph roughly a mirror image of the right half. The histogram on the next slide illustrates this.

Critical Thinking Interpreting Histograms Bell shape: 55 65 75 85 95 Example 1: What is the class width? What are the approximate lower and upper class limits of the first class?

The histogram below represents the number of television sets per household for a sample of U.S. households. Example 2: How many households are included in the sample? Example 3: How many households have 4 televisions? Example 4: What is the maximum number of households that have the same number of television sets?

Example 5: In a survey, 20 people were asked how many magazines they had purchased during the previous year. The results are shown below. Construct a histogram to represent the data. Use 4 classes, with a class width of 10. Does the distribution appear to be normal? 6 15 3 36 25 18 12 18 5 30 24 7 0 22 33 24 19 4 12 9 Intervals Frequency 0 – 9 7 10 – 19 6 20 – 29 4 30 – 39 3

Example 5: In a survey, 20 people were asked how many magazines they had purchased during the previous year. The results are shown below. Construct a histogram to represent the data. Use 4 classes, with a class width of 10. Does the distribution appear to be normal? Intervals Frequency 0 – 9 7 10 – 19 6 20 – 29 4 30 – 39 3

Histograms on the Calculator Press the STAT button. Choose “1. Edit” Enter your data in L1 Press “2nd Y=”(STAT PLOT) Choose “1: Plot 1…” Make your screen look like this: (turn it ON and choose histogram) Press “Zoom” Choose “9: ZoomStat” Press “Trace” and use the arrow buttons to move the cursor around your Histogram to determine class widths and frequencies.

Why do it? Histograms can make it easier to present information visually to someone that struggles with math They can help us visually determine our center They can also tell us more about the shape of our data…

Normal Distribution A very important concept in statistics is the normal distribution. Data distributions that are approximately normal have the following characteristics Start low Reach a maximum in the middle End low Roughly symmetric bell shape

Examples

Let’s Practice! Please work on Pg. 58 #5-8. In addition, go to Appendix B and find the full data set for the “Freshman 15” myth study. Make two histograms one of the students’ initial weights and one of the students’ ending weights. You make work by yourself, or with one to two other people.

In-Class Practice - #5 Part A: Approximately 60 cars. Part B: Approximately 20 cars.

In-Class Practice - #6 Part A: Approximately 5,000 miles. Part B: Approximately from 2,500 miles to 7,500 miles.

In-Class Practice - #7 Part A: Approximately 2,500 miles. Part B: Approximately from 42,500 miles.

In-Class Practice - #8 One possibility: The histogram includes mileage amounts from samples drawn from two different populations, such as the population of privately owned passenger cars and the population of taxi cabs, which are driven much longer distances.

Class width = 10. Minimum lower class limit = 35. In-Class Practice - #9 Class width = 10. Minimum lower class limit = 35.

Class width = 10. Minimum lower class limit = 35. In-Class Practice - #10 Class width = 10. Minimum lower class limit = 35.

Histograms Section 2.3 – Day 2

Why do it? Histograms can make it easier to present information visually to someone that struggles with math They can help us visually determine our center They can also tell us more about the shape of our data…

Normal Distribution A very important concept in statistics is the normal distribution. Data distributions that are approximately normal have the following characteristics Start low Reach a maximum in the middle End low Roughly symmetric bell shape

Center The center of normally distributed data is usually where you reach your maximum frequency. As you move left or right of the center, the frequency should drop the further away you get

Center The center can give us a lot of information about the data. For example…

Homework P.53: 13, 14