Histogram Distribution

Chapter 2

[2.1, 2.2]

STAT 109 MSU SPRING 2022

2.1 Visualizing Variation with Numerical Data

Distribution: States the values included in a data set and the frequency of each value within the data set. There are several types of distributions, including

- Dotplots

- Frequency Distributions / Histograms

- Relative Frequency Distributions​

2.1 Visualizing Variation with Numerical Data

Dotplots

In a dotplot, the values included in a data set are represented along the x-axis of a graph, and the frequency of each value is represented by the number of dots stacked above that value. There is one dot for each occurrence of that value in the data set.

2.1 Visualizing Variation with Numerical Data

Histograms

A frequency distribution, also called a histogram, is similar to a dot plot in that the values included in the data set are represented along the x-axis. In this distribution, the x-axis is separated into bins which cover a small range of values in the data set. There is a bar associated with each bin, with the height of each bar representing the frequency of data that falls in that bin within the data set. Frequency is represented on the y-axis.

2.1 Visualizing Variation with Numerical Data

Histograms

Changing the width of the bins in a histogram will change its shape. The bin width of a histogram should allow for important features to be communicated effectively.

​

A bin width that is too small can make the histogram confusing, and make trends in the data harder to spot.​

2.1 Visualizing Variation with Numerical Data

Histograms

Changing the width of the bins in a histogram will change its shape. The bin width of a histogram should allow for important features to be communicated effectively.

​

A bin width that is too large can make oversimplify the histogram, and fail to show any trends at all.

2.1 Visualizing Variation with Numerical Data

Relative Frequency Dist.

A relative frequency distribution is very similar to histogram. Instead of representing frequency on the y-axis, we represent relative frequency on the y-axis.

​

The shape of a histogram and a relative frequency distribution for the same data (with the same bin width) will always have exactly the same shape.​

2.1 Visualizing Variation with Numerical Data

Multiple Choice

The Internet Legal Research group website provides the bar exam pass rate of 196 law schools in the U.S. in 2013.

Approximately how many law schools had a pass rate between 90% and 100%?

• 75

• 10

• 35

• 5

Multiple Choice

The Internet Legal Research group website provides the bar exam pass rate of 196 law schools in the U.S. in 2013.

Approximately how many law schools had a pass rate above 80%

• 75

• 115

• 35

• 50

Multiple Choice

The Internet Legal Research group website provides the bar exam pass rate of 196 law schools in the U.S. in 2013.

In which bin is the highest pass rate of any law school represented in the histogram?

• 70%-80%

• 80%-90%

• 90%-100%

• 100%-110%

Multiple Choice

The Internet Legal Research group website provides the bar exam pass rate of 196 law schools in the U.S. in 2013.

Which range of pass rates has the highest frequency in this data set?

• 70%-80%

• 80%-90%

• 90%-100%

• 100%-110%

2.2 Summarizing Important Features of a Numerical Distribution

There are three features of utmost importance when analyzing a distribution.

- Shape

- Center

- Spread​

2.2 Summarizing Important Features of a Numerical Distribution

Shape (Symmetry and Skew)

- Symmetric Distributions: Equal amounts of data are represented on the left and right sides of the distribution. The distribution appears balanced and can be split in half down the middle, with the left and right sides matching perfectly.

2.2 Summarizing Important Features of a Numerical Distribution

Shape (Symmetry and Skew)

- Skewed Distributions have more data on one side and a tail of data spread out on the other.

- If the tail is on the right, then it is a right-skewed distribution. If the tail is on the left, then it is a left-skewed distribution.​

2.2 Summarizing Important Features of a Numerical Distribution

Multiple Choice

Which of these best describes the shape of the distribution?

• Left-Skewed

• Symmetric

• Right-Skewed

Multiple Choice

Which of these best describes the shape of the distribution?

• Left-Skewed

• Symmetric

• Right-Skewed

Shape (Modality)

- A mode is a value or range of values t​hat has a significantly higher frequency than neighboring values. This forms "peaks" in the distribution.

- A distribution with one mode is called unimodal. A distribution with two modes is called bimodal. A distribution with three or more modes is called multimodal.​

2.2 Summarizing Important Features of a Numerical Distribution

Multiple Choice

Which of these best describes the shape of the distribution?

• Unimodal

• Bimodal

• Multimodal

Multiple Choice

Which of these best describes the shape of the distribution?

• Unimodal

• Bimodal

• Multimodal

Center

- The center of a distribution can be represented by whatever value(s) or range of values occurs the most frequently. You might say that these values are "typical" in the data set. Thus, the center of a distribution could be represented by the modes of that data set. Visually, we look for peaks in the graph.

2.2 Summarizing Important Features of a Numerical Distribution

Fill in the Blank

Approximately what value represents the most typical value in this data set (the center)?

Fill in the Blank

Approximately what value represents the most typical value in this data set (the center)?

Center

Unfortunately, while it is easy to identify peaks in a distribution and say that these values are typical, these values often do a poor job of representing what is most typical in a dataset.

- When a distribution is heavily skewed, the peak in the graph is pushed to one side, and doesn't appear to be a "center" at all.

- When a distribution is multimodal, there is more than one typical value according to modality, so it doesn't make sense to call any of those values the "center."

- Using modality to identify typical values is a short cut that we use sparingly.​

2.2 Summarizing Important Features of a Numerical Distribution

Spread

- The spread of a distribution is a representation of the variability within the data set.

- Distributions with a high degree of variation will be "spread" out. The histogram will cover a large range of values and the peaks in the distribution may not be very significant.

​- Distributions with a low degree of variability will have their data lumped very close together. Peaks in the histogram will likely be extreme.

2.2 Summarizing Important Features of a Numerical Distribution

Comparisons

Summarizing these features across multiple distributions allows us to compare those distributions objectively, and make inferences about the relevant populations. These comparisons and subsequent inferences are the basis for further statistical testing to be performed in formal studies.

2.2 Summarizing Important Features of a Numerical Distribution

Multiple Choice

The histograms represent the distributions of student-to-teacher ratios at many randomly selected private and public colleges in the United States. Select the most accurate comparison.

• There is a larger proportion of private schools in the study with a low student-to-teacher ratio than there is for public schools.

• There is a larger proportion of public schools in the study with a low student-to-teacher ratio than there is for private schools.

• The proportion of schools with low student-to-teacher ratios is roughly the same in both distributions.

Multiple Choice

The histograms represent the distributions of student-to-teacher ratios at many randomly selected private and public colleges in the United States. Select the most accurate comparison.

• It is typical for private schools in this study to have a lower student-to-teacher ratio than public schools in this study.

• It is typical for public schools in this study to have a lower student-to-teacher ratio than private schools in this study.

• Private and public schools in this study typically have the same or similar student-to-teacher ratios.

Multiple Select

The histograms represent the distributions of student-to-teacher ratios at many randomly selected private and public colleges in the United States. Select the most accurate comparison.

• There is more variability in student-to-teacher ratios at private schools in this study than there is at public schools in this study.

• There is more variability in student-to-teacher ratios at public schools in this study than there is at private schools in this study.

• Variability in student-to-teacher ratios is the same or similar at private and public schools in this study.

Comparisons

Consider the comparisons you made in the last three questions. Discuss with a partner or small group: What inference can be made about the relevant populations from which these schools were sampled?

2.2 Summarizing Important Features of a Numerical Distribution