Step-by-Step Guidance for Introductory Statistics Worksheet: Histograms and Distribution Analysis

Q1. Draw a histogram of the number of points scored per game. Does the number of points scored per game appear to have a normal distribution?

Background

Topic: Descriptive Statistics & Distribution Analysis

This question tests your ability to construct a histogram from raw data and interpret the shape of the distribution, specifically considering whether it resembles a normal distribution.

Key Terms and Concepts:

• Histogram: A graphical representation of the distribution of numerical data, where the data is divided into bins (intervals) and the frequency of data in each bin is shown as a bar.

• Normal Distribution: A symmetric, bell-shaped distribution characterized by most values clustering around the mean, with fewer values at the extremes.

Step-by-Step Guidance

1. List all the points scored in each game: 8, 15, 15, 12, 13, 10, 19, 22, 22, 28, 20, 16, 16, 10, 14, 14, 7, 9.

2. Determine appropriate bin intervals for the histogram. For example, you might use bins such as 5–9, 10–14, 15–19, 20–24, 25–29.

3. Count how many games fall into each bin interval.

4. Draw the histogram, labeling the x-axis with the bin intervals and the y-axis with the frequency (number of games in each bin).

5. Examine the shape of the histogram. Consider whether it is symmetric, skewed, or has multiple peaks.

Try solving on your own before revealing the answer!

Final Answer:

The histogram shows that the distribution of points scored per game is not perfectly normal. The data is somewhat skewed to the left, with more games having lower scores and a few games with higher scores. There is no clear bell-shaped curve, so the distribution does not appear to be normal.

Q2a. Graph the diastolic blood pressure of these patients beginning with a lower class limit of 40 mm Hg and with a class width of 9 mm Hg. Does the data appear to be normally distributed?

Background

Topic: Frequency Distributions & Normality Assessment

This question asks you to create a histogram for diastolic blood pressure data and assess whether the distribution is normal, using StatCrunch and overlaying a normal curve for comparison.

Key Terms and Formulas:

• Class Limit: The lowest value that can be included in a class (bin).

• Class Width: The difference between the lower limits of consecutive classes.

• Overlay Distribution: Adding a normal curve to the histogram to visually compare the data distribution to a normal distribution.

Step-by-Step Guidance

1. Open the Body Data set in StatCrunch.

2. Create a histogram for diastolic blood pressure, setting the lower class limit to 40 mm Hg and the class width to 9 mm Hg.

3. Use the "Overlay Distribution: Normal" option to add a normal curve to your histogram.

4. Compare the shape of your histogram to the normal curve. Look for symmetry, bell shape, and whether the data clusters around the mean.

Try solving on your own before revealing the answer!

Final Answer:

With a class width of 9 mm Hg, the histogram may show some resemblance to a normal distribution, but there could be deviations such as skewness or outliers. The overlay helps visualize how closely the data matches a normal curve, but the distribution may not be perfectly normal.

Q2b. Make a new graph of the diastolic blood pressure of these patients again beginning with a lower class limit of 40 mm Hg and with a class width of 10 mm Hg. In what way does this graph differ from the graph in part (a)?

Background

Topic: Effect of Bin Width on Histograms

This question tests your understanding of how changing the class width (bin size) affects the appearance and interpretation of a histogram.

Key Terms:

• Class Width/Bin Size: The interval size for grouping data in a histogram.

Step-by-Step Guidance

1. Repeat the histogram creation process, but set the class width to 10 mm Hg instead of 9 mm Hg.

2. Observe how the number of bins changes and how the data is grouped differently.

3. Compare the new histogram to the previous one. Consider whether the shape is smoother or rougher, and if any features (such as peaks or gaps) are more or less apparent.

Try solving on your own before revealing the answer!

Final Answer:

With a class width of 10 mm Hg, the histogram will have fewer bins, which can make the distribution appear smoother and may hide some details compared to the 9 mm Hg width. The grouping of data changes, potentially affecting the visual interpretation of normality.

Q2c. Make a new graph of the diastolic blood pressure of these patients again beginning with a lower class limit of 40 mm Hg and with a class width of 15 mm Hg. In what way does this graph differ from the graph in part (a)?

Background

Topic: Histogram Interpretation & Bin Width Effects

This question further explores how increasing the class width affects the histogram and the visibility of distribution features.

Key Terms:

• Class Width: The interval size for grouping data in a histogram.

Step-by-Step Guidance

1. Create a histogram with a class width of 15 mm Hg, starting at 40 mm Hg.

2. Notice how the number of bins decreases further, and more data is grouped together in each bin.

3. Compare this histogram to the one with a 9 mm Hg class width. Consider how the shape and details of the distribution change.

Try solving on your own before revealing the answer!

Final Answer:

With a class width of 15 mm Hg, the histogram will have even fewer bins, which can obscure finer details and make the distribution appear less detailed. This can make it harder to assess normality or spot features like skewness or outliers.

Q3a. Create a histogram for shoe size grouped by sex. What differences in distribution do you notice between the two histograms?

Background

Topic: Comparative Distribution Analysis

This question asks you to compare the distributions of shoe sizes for different sexes using histograms.

Key Terms:

• Grouped Histogram: A histogram that displays data for different groups (e.g., male and female) side by side or stacked.

• Distribution: The way values are spread across the range of data.

Step-by-Step Guidance

1. Open the Foot and Height data set in StatCrunch.

2. Create histograms for shoe size, using the "Group by" option to separate by sex.

3. Stack the histograms vertically by setting "Rows per Page" to 2.

4. Compare the two histograms. Look for differences in central tendency, spread, and shape.

Try solving on your own before revealing the answer!

Final Answer:

The histograms show that one sex (typically males) tends to have larger shoe sizes, with the distribution shifted to the right compared to the other group. The spread and shape may also differ, with one group having a wider range of sizes.

Q3b. Now edit the graph and mark “Use the same x-axis” in the “For multiple graphs” section. What difference in distribution is now more apparent between the two histograms? Why was it deceptive to stack the two graphs with different x-axes?

Background

Topic: Data Visualization & Axis Consistency

This question tests your understanding of how axis scaling affects the interpretation of comparative histograms.

Key Terms:

• x-axis: The horizontal axis representing the variable (shoe size).

• Axis Consistency: Using the same scale for comparison across groups.

Step-by-Step Guidance

1. Edit the histogram settings to use the same x-axis for both groups.

2. Observe how the distributions align and compare more directly.

3. Consider why using different x-axes could mislead viewers about the differences between groups.

Try solving on your own before revealing the answer!

Final Answer:

Using the same x-axis makes the difference in shoe size distributions more apparent, showing that one group has larger sizes overall. Stacking graphs with different x-axes can be deceptive because it may exaggerate or hide differences, making comparisons inaccurate.

Q3c. Find the units for SHOE SIZE in Appendix B of the book. Is there any reason to think there may be bias in the data?

Background

Topic: Data Collection & Bias

This question asks you to consider the units used for shoe size and whether the data collection method could introduce bias.

Key Terms:

• Units: The measurement system used (e.g., US, UK, EU shoe sizes).

• Bias: Systematic error introduced by the data collection process.

Step-by-Step Guidance

1. Locate Appendix B in your textbook to find the units for shoe size.

2. Consider whether the units are consistent and appropriate for all participants.

3. Think about whether the sample might be biased (e.g., only includes certain age groups, regions, or types of shoes).

Try solving on your own before revealing the answer!

Final Answer:

If the units are not standardized or if the sample is not representative (e.g., only US sizes, only college students), there may be bias in the data. This could affect the validity of conclusions drawn from the data.

Q4. Using the Fast Food Data set, create histograms on StatCrunch for all four fast food companies for lunch stacked vertically with the same x-axis. Should the bars all be the same width?

Background

Topic: Histogram Construction & Comparison

This question tests your understanding of how to construct comparative histograms and the importance of consistent bar width for accurate comparison.

Key Terms:

• Bar Width: The width of each bin in a histogram.

• Comparative Histograms: Histograms for multiple groups displayed together for comparison.

Step-by-Step Guidance

1. Open the Fast Food Data set in StatCrunch.

2. Create histograms for each company, stacking them vertically and using the same x-axis.

3. Consider whether the bars (bins) should have the same width for all companies to ensure fair comparison.

Try solving on your own before revealing the answer!

Final Answer:

Yes, the bars should all be the same width to allow for accurate comparison between companies. Different bar widths can distort the visual representation and make comparisons misleading.