Lesson 2 Homework Practice Histograms Answer Key

Lesson 2 Homework – Practice Histograms: Answer Key and How to Master the Concepts

Histograms are one of the most powerful tools for visualizing the distribution of a data set, and Lesson 2 in most introductory statistics courses focuses on building, interpreting, and critiquing them. This answer key not only provides the correct solutions for the homework problems but also explains the reasoning behind each step, so you can confidently tackle any histogram question on exams or future projects.

Introduction: Why Histograms Matter

A histogram groups continuous data into bins (or class intervals) and displays the frequency—or relative frequency—of observations in each bin as adjacent bars. Unlike bar charts, the bars in a histogram touch each other, emphasizing that the data are continuous. Mastery of histograms enables you to:

• Detect skewness, modality, and outliers at a glance.
• Compare multiple distributions side‑by‑side.
• Summarize large data sets without listing every single value.

The following sections walk through each homework problem, present the correct answer, and highlight the statistical concepts you should retain.

Problem 1 – Constructing a Histogram

Data set (n = 20):
12, 15, 17, 19, 22, 23, 24, 24, 26, 27, 28, 30, 31, 33, 35, 36, 38, 40, 42, 45

Task: Choose an appropriate bin width, create the frequency table, and draw the histogram.

Answer Key

1. Determine the range
   [ \text{Range}= \max - \min = 45 - 12 = 33 ]

2. Select the number of bins – A common rule of thumb is Sturges’ formula:
   [ k = 1 + \log_2(n) \approx 1 + \log_2(20) \approx 5.3 \rightarrow 5\text{ or }6\text{ bins} ]

3. Calculate bin width (rounded to a convenient number)
   [ \text{Width}= \frac{\text{Range}}{k}= \frac{33}{6}=5.5 \approx 6 ]

4. Define bin edges (starting at the minimum, 12)

5. Draw the histogram – Plot the bins on the horizontal axis and frequencies on the vertical axis; use bars of width 6 that touch each other.

Interpretation: The distribution is roughly symmetrical with a slight right‑hand tail (the 42–47 bin). No extreme outliers appear.

Problem 2 – Relative Frequency Histogram

Using the same data set, convert frequencies to relative frequencies (percentages) and redraw the histogram. The details matter here.

Answer Key

Relative frequency for each bin = (\frac{\text{Frequency}}{n} \times 100%).

If you're redraw the histogram, the vertical axis now shows percentages instead of raw counts. The visual shape remains identical, reinforcing that relative frequency histograms preserve the same pattern while allowing comparison across data sets of different sizes.

Problem 3 – Identifying the Mode from a Histogram

A histogram (not shown here) displays the following frequencies for bins of exam scores:

Question: What is the modal class, and what approximate value would you report as the mode?

Answer Key

• Modal class = the bin with the highest frequency → 70‑79.
• Approximate mode – For a grouped data set, a common estimate is the midpoint of the modal class:
  [ \text{Midpoint}= \frac{70+79}{2}=74.5 ]

Hence, the mode is approximately 74.5 points.

Concept check: The mode is a measure of central tendency that reflects the most common interval, not a precise single value.

Problem 4 – Comparing Two Histograms

You are given two histograms representing the heights (in cm) of male and female students.

Male histogram: peaks around 175 cm, slight right skew.
Female histogram: peaks around 162 cm, fairly symmetric.

Task: List three statistical observations that can be made from the side‑by‑side comparison.

Answer Key

1. Location difference – The male distribution’s peak (mean/median) is higher than the female’s, indicating that on average males are taller.
2. Shape contrast – The male histogram’s right skew suggests a few exceptionally tall individuals, whereas the female histogram’s symmetry implies a more balanced spread.
3. Spread (variability) – The male histogram appears wider, hinting at a larger standard deviation; the female histogram is narrower, indicating less variability in height.

These observations would be confirmed by calculating descriptive statistics (mean, median, SD) for each group.

Problem 5 – Cumulative Frequency Histogram (Ogive)

Given the frequency table from Problem 1, construct a less‑than‑or‑equal‑to (≤) ogive and determine the approximate 75th percentile (Q3).

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Answer Key

1. Cumulative frequencies

2. Plot the ogive – Horizontal axis = upper bound of each bin; vertical axis = cumulative frequency. Connect points with straight lines.

3. Locate the 75th percentile – 75% of 20 observations = 15. Find where cumulative frequency reaches 15.

   • Between the 35‑bin (cumulative 16) and the 29‑bin (cumulative 12).
   • Interpolate linearly:

   [ Q_3 \approx 29 + \frac{15-12}{16-12}\times(35-29) = 29 + \frac{3}{4}\times6 = 29 + 4.5 = 33.5 ]

Thus, the 75th percentile is roughly 33.5 cm (or the corresponding measurement unit used in the original data).

Takeaway: Ogives are excellent for estimating quantiles without performing detailed calculations on raw data.

Problem 6 – Choosing the Right Bin Width

A student creates a histogram of 200 daily temperatures using 30 bins and finds the plot looks “spiky.” The instructor suggests reducing the number of bins.

Question: Explain why too many bins can be problematic and recommend a better bin count using the Rice Rule.

Answer Key

Too many bins → each bin contains few observations, causing random fluctuations to dominate and obscuring the underlying shape (over‑fitting).

Rice Rule:
[ k = 2 \times n^{1/3} ]
where (n = 200).

[ k = 2 \times 200^{1/3} \approx 2 \times 5.85 \approx 11.7 \rightarrow 12 \text{ bins} ]

So, 12 – 13 bins would give a smoother, more interpretable histogram.

Conceptual note: The ideal bin width balances bias (too few bins) and variance (too many bins).

Problem 7 – Interpreting a Skewed Histogram

A histogram of household incomes shows a long right tail, with most bars clustered between $30k and $60k, and a few bars extending beyond $200k.

Task: Identify the type of skewness, suggest the most appropriate measure of central tendency, and propose a transformation to normalize the data.

Answer Key

1. Skewness: Right‑skewed (positively skewed).
2. Best central tendency: Median (or the trimmed mean) because the mean is pulled upward by the extreme high incomes.
3. Normalization transformation: Apply a logarithmic transformation (e.g., ( \log_{10}(\text{income}))). This compresses large values, often turning a right‑skewed distribution into an approximately symmetric one, making parametric analyses more valid.

Frequently Asked Questions (FAQ)

Q1: Can I use unequal bin widths?

A: Yes, especially when data are sparse in certain ranges. When bins differ in width, you must plot density (frequency divided by bin width) on the vertical axis so the area of each bar still reflects the proportion of observations.

Q2: What’s the difference between a histogram and a bar chart?

A: A histogram visualizes continuous data with touching bars; a bar chart displays categorical data with gaps between bars. The y‑axis of a histogram represents frequency (or density), whereas a bar chart can represent any categorical count.

Q3: How do I decide whether to display frequencies or relative frequencies?

A: Use relative frequencies (percentages) when comparing distributions of different sample sizes, or when you want to underline the proportion of observations rather than raw counts.

Q4: Is there a rule for the “optimal” number of bins?

A: No single rule works for every data set, but common guidelines include Sturges’ formula, Rice Rule, Scott’s normal reference rule, and Freedman‑Diaconis rule. Try a few and choose the one that reveals the underlying pattern without over‑complicating the plot.

Q5: Why do some textbooks recommend a minimum of five bins?

A: Fewer than five bins usually hide essential features such as modality or skewness. Five to ten bins often provide enough granularity for most moderate‑size data sets.

Conclusion – Turning Practice Into Mastery

The Lesson 2 homework practice histograms answer key does more than give you the right numbers; it walks you through the why behind each decision—choosing bin width, interpreting shape, estimating quantiles, and handling skewness. By internalizing these concepts:

• You will read histograms quickly, spotting key features without hesitation.
• You will create accurate, aesthetically clean histograms that convey the story of your data.
• You will be prepared for advanced topics such as kernel density estimation and probability distribution fitting.

Remember, the best way to cement these skills is to re‑create the plots using software (Excel, R, Python’s Matplotlib/Seaborn) and then explain the findings aloud as if teaching a peer. The more you practice, the more intuitive the decisions become, and the more confident you’ll feel when a professor or a manager asks you to “show the distribution.”

Happy histogramming!