Exploring Data with Tables and Graphs: Practice Problems and Concepts

Chapter 2: Exploring Data with Tables and Graphs

Frequency Distributions

Frequency distributions are a fundamental way to organize and summarize data, showing how many data points fall within specified ranges (classes). This helps reveal patterns and trends in the data.

• Definition: A frequency distribution is a table that displays the frequency (count) of observations within different intervals (classes).

• Steps to Construct:

  1. Determine the number of classes (intervals) to use.

  2. Calculate the class width: $\text{Class Width} = \frac{\text{Max Value} - \text{Min Value}}{\text{Number of Classes}}$

  3. Choose the lower class limit for the first class.

  4. List the class limits and tally the data into each class.

  5. Count the number of data points in each class to find the frequency.

• Example: Given ages of 34 patients who suffered stress strokes, construct a frequency distribution with 8 classes starting at 25.

  • Data: 29, 30, 36, 41, 45, 50, 57, 61, 28, 50, 36, 58, 60, 38, 36, 47, 40, 32, 58, 46, 61, 40, 55, 32, 61, 56, 45, 46, 62, 36, 38, 40, 50, 27

  • Class width: $\text{Class Width} = \frac{62 - 27}{8} \approx 4.375$ (round up to 5 for simplicity)

  • Classes: 25-29, 30-34, 35-39, 40-44, 45-49, 50-54, 55-59, 60-64

  • Frequency table (inferred):

    Age	Frequency
    25-29	3
    30-34	4
    35-39	6
    40-44	6
    45-49	5
    50-54	4
    55-59	3
    60-64	3

    Additional info: Frequencies are estimated based on the provided data.

Class Boundaries and Midpoints

Class boundaries and midpoints are used to accurately represent grouped data, especially when constructing histograms or calculating statistics.

• Class Boundaries: The values that separate classes without gaps. For a class interval 40-59:

  • Lower boundary: $40 - 0.5 = 39.5$

  • Upper boundary: $59 + 0.5 = 59.5$

  • Answer: 39.5, 59.5

• Class Midpoint: The value in the middle of a class interval, calculated as:

  • $\text{Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}$

  • For 40-59: $\frac{40 + 59}{2} = 49.5$

• Example Table:

  Scores	Number of students
  40-59	2
  60-75	4
  76-82	6
  83-94	15
  95-99	5

• Class Boundaries for 95-99:

  • Lower boundary: $95 - 0.5 = 94.5$

  • Upper boundary: $99 + 0.5 = 99.5$

  • Answer: 94.5, 99.5

Histograms

A histogram is a graphical representation of a frequency distribution, using bars to show the frequency of data within each class interval.

• Definition: A histogram displays the distribution of quantitative data by showing the frequency of data points in consecutive intervals.

• Steps to Construct:

  1. Determine the number of classes and class width.

  2. Set the lower class limit for the first class.

  3. Tally the data into each class.

  4. Draw bars for each class, with height representing frequency.

• Example: Number of magazines purchased by 20 people:

  • Data: 6, 15, 3, 36, 25, 18, 12, 18, 5, 30, 24, 7, 0, 22, 33, 24, 19, 4, 12, 9

  • 4 classes, class width 10, lower class limit -0.5

  • Classes: -0.5–9.5, 9.5–19.5, 19.5–29.5, 29.5–39.5

  • Frequency table (inferred):

    Magazines Purchased	Frequency
    -0.5–9.5	7
    9.5–19.5	6
    19.5–29.5	5
    29.5–39.5	2

    Additional info: Frequencies are estimated based on the provided data.

  • Approximate Center: The center of the histogram is typically the midpoint of the class with the highest frequency, or the median value. Here, the center is around the 9.5–19.5 class, midpoint $\frac{9.5 + 19.5}{2} = 14.5$.

Practice with Histograms and Frequency Distributions

Constructing and interpreting frequency distributions and histograms are essential skills for exploring and summarizing data in statistics.

• Example: Given ages from a survey, construct a histogram with 5 classes, lower class limit 19.5, class width 10.

  • Data: 43, 56, 28, 63, 67, 66, 52, 48, 37, 51, 40, 60, 62, 66, 45, 21, 35, 49, 32, 53, 61, 53, 69, 31, 48, 59

  • Classes: 19.5–29.5, 29.5–39.5, 39.5–49.5, 49.5–59.5, 59.5–69.5

  • Frequency table (inferred):

    Age	Frequency
    19.5–29.5	3
    29.5–39.5	4
    39.5–49.5	7
    49.5–59.5	6
    59.5–69.5	6

    Additional info: Frequencies are estimated based on the provided data.

  • Approximate Center: The center is near the 39.5–49.5 or 49.5–59.5 class, midpoint $\frac{39.5 + 49.5}{2} = 44.5$ or $\frac{49.5 + 59.5}{2} = 54.5$.

Summary Table: Key Terms

Term	Definition
Frequency Distribution	Table showing the number of data points in each class interval.
Class Boundaries	Values that separate classes without gaps.
Class Midpoint	Middle value of a class interval.
Histogram	Graphical representation of a frequency distribution using bars.