Histograms and Data Grouping: Constructing a 5-Class Histogram

Histograms in Statistics

Definition and Purpose

A histogram is a graphical representation of the distribution of numerical data. It is used to visualize the frequency of data values within specified intervals, known as classes or bins. Histograms are particularly useful for understanding the shape, spread, and central tendency of a dataset.

• Key Terms:

  • Class Interval (Bin): A range of values into which data is grouped.

  • Frequency: The number of data points falling within a class interval.

• Application: Histograms are commonly used in statistics to summarize large datasets and identify patterns such as skewness, modality, and outliers.

Constructing a Histogram: Step-by-Step

Step 1: Organize the Data

The provided data represents the number of pushups completed by 7th graders:

• Raw Data: 5, 8, 6, 10, 12, 22, 29, 6, 7, 8, 9, 19, 11, 11, 11, 12, 11, 16, 18, 12, 15, 14, 16, 20, 21, 20, 16, 20, 17, 13, 15, 14, 18, 20, 11, 19, 21, 22, 7, 9, 4

Step 2: Determine the Number of Classes

• The question asks for a 5-class histogram.

Step 3: Find the Range and Class Width

• Minimum value: 4

• Maximum value: 29

• Range: $29 - 4 = 25$

• Class Width: $\text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}} = \frac{25}{5} = 5$

• To ensure all data is included, round up if necessary. Here, class width of 5 is appropriate.

Step 4: Define Class Intervals

• Start at the minimum value (4), add the class width (5) to define each interval:

• Class 1: 4 - 8

• Class 2: 9 - 13

• Class 3: 14 - 18

• Class 4: 19 - 23

• Class 5: 24 - 28 (but since max is 29, extend to 29)

• Final Intervals:

  • 4 - 8

  • 9 - 13

  • 14 - 18

  • 19 - 23

  • 24 - 29

Step 5: Tally Frequencies

Class Interval	Frequency
4 - 8	8
9 - 13	11
14 - 18	10
19 - 23	8
24 - 29	1

Additional info: Frequencies were counted by grouping each data point into its respective interval.

Step 6: Draw the Histogram

• On the x-axis, plot the class intervals.

• On the y-axis, plot the frequency for each interval.

• Each bar's height corresponds to the frequency of the interval.

• Note: Since this is a text-based guide, the histogram is described rather than drawn.

Interpreting the Histogram

• The histogram shows the distribution of pushup counts among 7th graders.

• Most students completed between 9 and 18 pushups.

• Very few students completed more than 24 pushups.

• This distribution can help identify the overall fitness level of the class.

Example: Histogram Application

• Example: If a school wants to set a benchmark for physical fitness, they can use the histogram to determine a reasonable target based on the most common performance range.

Summary Table: Histogram Construction Steps

Step	Description
1	Organize raw data
2	Determine number of classes
3	Calculate range and class width
4	Define class intervals
5	Tally frequencies
6	Draw and interpret histogram