A histogram shows how often data values fall within intervals, making it one of the fastest ways to understand a data set. The overall shape can reveal patterns that are hidden in a list of numbers. Symmetric, skewed, uniform, and bimodal shapes each suggest different stories about the data.
Recognizing these shapes helps students choose better summaries and make better comparisons.
Key Facts
- A symmetric distribution has roughly matching left and right sides, and the mean is usually close to the median.
- A right-skewed distribution has a long tail to the right, and often mean > median.
- A left-skewed distribution has a long tail to the left, and often mean < median.
- A uniform distribution has bars with nearly equal heights, meaning values occur at similar frequencies across the range.
- A bimodal distribution has two clear peaks, which may suggest two different groups or processes in the data.
- Relative frequency = class frequency / total number of observations.
Vocabulary
- Histogram
- A graph that displays the frequency of numerical data values within intervals called bins.
- Skewness
- Skewness describes how much a distribution has a longer tail on one side than the other.
- Symmetric distribution
- A symmetric distribution has a shape where the left and right sides are approximately mirror images.
- Bimodal distribution
- A bimodal distribution has two distinct peaks, showing that two value ranges occur especially often.
- Outlier
- An outlier is a data value that is far from most other values in the data set.
Common Mistakes to Avoid
- Calling any tall bar a mode, which is wrong because a mode is a value or interval with especially high frequency compared with nearby intervals.
- Ignoring the tail when naming skew, which is wrong because skew is named for the direction of the long tail, not the side with the tallest bars.
- Using the mean alone for a strongly skewed distribution, which is wrong because extreme values can pull the mean away from the typical data value.
- Assuming a bimodal histogram is just random noise, which is wrong because two peaks can point to two separate groups, conditions, or processes.
Practice Questions
- 1 A histogram of quiz scores has bin frequencies 2, 5, 9, 5, 2 from lowest to highest score bins. What shape is suggested, and should the mean be close to the median?
- 2 A data set has 40 observations. In one histogram bin, the frequency is 10. What is the relative frequency for that bin?
- 3 A histogram of commute times has many values between 10 and 25 minutes, but a few values stretch out to 80 minutes. Identify the likely shape and explain which measure of center, mean or median, better represents a typical commute.