Bar Graphs and Histograms - Sixth Grade Mathematics

Bar graphs and histograms use rectangular bars to represent frequencies, but serve fundamentally different roles based on categorical versus continuous numerical intervals.

The Critical Difference: Bar Graphs vs. Histograms

Although they look superficially similar, confusing bar graphs and histograms is one of the most common errors in introductory statistics.

+-----------------------+-----------------------------+-----------------------------+
| Feature               | Bar Graph                   | Histogram                   |
+-----------------------+-----------------------------+-----------------------------+
| Type of Data          | Categorical (Qualitative)   | Continuous Numerical Data   |
+-----------------------+-----------------------------+-----------------------------+
| X-Axis Representation | Distinct categories/labels  | Consecutive numerical bins  |
+-----------------------+-----------------------------+-----------------------------+
| Spacing Between Bars  | Spaces / Gaps between bars  | NO gaps between bars        |
|                       | (shows separation)          | (bars touch to show continuity)
+-----------------------+-----------------------------+-----------------------------+
| Order of Bars         | Arbitrary (alphabetical,    | Strict numerical order from |
|                       | frequency, preference)      | lowest bin to highest bin   |
+-----------------------+-----------------------------+-----------------------------+

Visual Comparison:
Bar Graph (Spaces Between Categories):
    Freq |
      10 |      +---+
       8 |      |   |       +---+
       6 |  +---+   |       |   |
       4 |  |   |   |   +---+   |
         +--+---+---+---+---+---+--->
             Cat   Dog  Bird Fish  (Categories)

Histogram (Bars Touch Across Equal Intervals):
    Freq |
      10 |          +---+
       8 |          |   |---+
       6 |      +---+   |   |
       4 |  +---+   |   |   |---+
         +--+---+---+---+---+---+--->
            0-9 10-19 20-29 30-39 40-49 (Equal Bins)

In a histogram, the bars touch because the horizontal axis is a continuous number line. A gap only occurs if a particular bin interval contains zero observations.

Setting Up Intervals (Bins) for a Histogram

To construct a histogram:
1. Choose an interval width (bin size). All intervals must have the exact same width!

2. Intervals must not overlap (e.g., 0-9, 10-19, 20-29; or 0 to <10, 10 to <20).

3. Every data point must fall into exactly one bin.

Creating Intervals from Raw Data (Daily Study Times in Minutes for 25 Students):
Raw Data: 12, 18, 25, 29, 30, 32, 35, 40, 44, 45, 48, 50, 52, 55, 58, 60, 62, 65, 70, 75, 80, 85, 88, 90, 95

Grouped Frequency Table (Bin Width = 20 minutes):
+---------------------+--------------+-----------+
| Interval (Minutes)  | Tally        | Frequency |
+---------------------+--------------+-----------+
| 0 to 19             | ||           | 2         |
| 20 to 39            | |||| |       | 5         |
| 40 to 59            | |||| |||     | 8         |
| 60 to 79            | |||| |       | 5         |
| 80 to 99            | ||||         | 5         |
+---------------------+--------------+-----------+

Interpreting Histograms

When interpreting a histogram, examine:
1. Shape: Is it symmetric (bell-shaped), skewed right (longer tail on high values), or skewed left?

2. Peak: Which interval contains the highest bar (modal class)?

3. Total Sample Size: Found by summing the heights of all the bars.

Notice that from a histogram alone, you cannot identify individual original data values. If a bar shows that 8 students studied between 40 and 59 minutes, you know how many students fall into that range, but you cannot tell whether someone studied exactly 45 minutes or 52 minutes.

Exercises

Exercise 1. State whether a bar graph or a histogram is the appropriate choice to display: (a) The number of students who walk, ride a bus, ride a bike, or take a car to school; (b) The distribution of marathon finishing times for 500 runners; (c) The sales of four different flavors of juice; (d) The test percentages of 100 students grouped into 10% ranges.

Exercise 2. Given the histogram frequency distribution: Interval 10-19: frequency 4 Interval 20-29: frequency 7 Interval 30-39: frequency 12 Interval 40-49: frequency 6 Interval 50-59: frequency 1 (a) How many total observations are represented? (b) Which interval is the peak (mode)? (c) What percent of observations are 30 or greater? (d) Can you determine the exact highest value in the data set? Explain.

Exercise 3. Explain why the bars in a histogram must touch, whereas the bars in a bar graph should have spaces between them.

Solutions

Solution to Exercise 1. For (a), a bar graph is appropriate because transportation modes are discrete categories. For (b), a histogram is appropriate because marathon finishing times are continuous numerical measurements grouped into intervals. For (c), a bar graph is appropriate because juice flavors are categorical. For (d), a histogram is appropriate because test percentages are continuous numerical scores grouped into equal numerical bins.

Solution to Exercise 2. For (a), sum the frequencies: 4 + 7 + 12 + 6 + 1 = 30 total observations. For (b), the peak is interval 30-39 with the highest frequency of 12. For (c), count observations in intervals 30-39, 40-49, and 50-59: 12 + 6 + 1 = 19. Percentage = 19 / 30 = approximately 63.3%. For (d), no, you cannot determine the exact highest value; you only know that one observation lies somewhere between 50 and 59.

Solution to Exercise 3. In a histogram, the horizontal axis represents a continuous numerical scale where the end of one bin interval connects directly to the start of the next bin interval, so the bars touch to illustrate this continuous flow of numbers. In a bar graph, the horizontal axis lists separate, independent qualitative categories with no mathematical continuum between them, so gaps are intentionally placed between the bars to visually emphasize their distinct categorical separation.