뒤로Histograms and Data Distributions in Introductory Statistics
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Section 2.2: Histograms
Definition and Construction of Histograms
A histogram is a graphical representation of the distribution of quantitative data. It consists of adjacent bars of equal width, where:
The horizontal axis represents quantitative data, using class boundaries, midpoints, or lower class limits from a frequency distribution.
The vertical axis shows the frequency (or relative frequency) of data within each class interval.
Histograms are useful for visualizing the shape, center, and spread of data, as well as identifying outliers.
Example: The histogram above displays McDonald's lunch service times (in seconds). Each bar's height shows how many observations fall within each time interval.
Notes About Histograms
You can create relative frequency histograms by using percentages instead of raw frequencies on the vertical axis.
Histograms are generally easier to interpret than frequency tables, as they provide a visual summary of the data's distribution.
Histograms help analyze key features of data, summarized by the acronym CVDOT:
Center (location of the middle of the data)
Variation (spread of the data)
Distribution (overall shape)
Outliers (unusual values)
Time (changes over time, if applicable)
Outliers
An outlier is a data value that is significantly different from the majority of the data set. Outliers can indicate variability in measurement, experimental errors, or novel findings.
Example: In the data set 150, 145, 160, 165, 350, 140, 130 pounds, the value 350 is an outlier because it is much higher than the other values.
Distributions
Definition of Distribution
The distribution of a data set describes the shape of how data values are spread over the range of possible values. Understanding the distribution helps in identifying patterns and making inferences about the population.
Common Types of Distributions
Normal Distribution: Also known as a bell-shaped curve, it is symmetric about the center and most data values cluster around the mean.
Uniform Distribution: All intervals have approximately the same frequency, resulting in a flat, rectangular shape.
Skewed to the Right (Positively Skewed): Most data are concentrated on the left, with a tail extending to the right.
Skewed to the Left (Negatively Skewed): Most data are concentrated on the right, with a tail extending to the left.
Example: Identifying Distribution Shape
Scenario: You give an easy test where most students score very high. When you plot the scores as a histogram, what type of distribution would you expect?
Solution: The distribution would be skewed to the left (negatively skewed), since most students score near 100% and only a few score much lower. This matches the shape shown in the fourth histogram above.
Summary Table: Types of Distributions
Distribution Type | Shape Description | Example |
|---|---|---|
Normal | Symmetric, bell-shaped | Heights of adult men |
Uniform | Flat, all intervals equally likely | Rolling a fair die |
Skewed Right | Tail on the right | Income distribution |
Skewed Left | Tail on the left | Easy test scores |
Key Formulas
Relative Frequency:
Class Midpoint:
