Learning Outcomes
- Recognize, describe, and calculate the measures of the center of data: mean, median, and mode.
Consider the following data set.
[latex]4[/latex]; [latex]5[/latex]; [latex]6[/latex]; [latex]6[/latex]; [latex]6[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]8[/latex]; [latex]8[/latex]; [latex]8[/latex]; [latex]9[/latex]; [latex]10[/latex]
This data set can be represented by following histogram. Each interval has width one, and each value is located in the middle of an interval.
The histogram for the data: [latex]4[/latex]; [latex]5[/latex]; [latex]6[/latex]; [latex]6[/latex]; [latex]6[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]8[/latex] is not symmetrical. The right-hand side seems “chopped off” compared to the left side. A distribution of this type is called skewed to the left because it is pulled out to the left.
The histogram for the data: [latex]6[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]8[/latex]; [latex]8[/latex]; [latex]8[/latex]; [latex]9[/latex]; [latex]10[/latex], is also not symmetrical. It is skewed to the right.
To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.
Skewness and symmetry become important when we discuss probability distributions in later chapters.
Here is a video that summarizes how the mean, median and mode can help us describe the skewness of a dataset. Don’t worry about the terms leptokurtic and platykurtic for this course.
Example
Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.
Terry: [latex]7[/latex]; [latex]9[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]4[/latex]; [latex]1[/latex]; [latex]3[/latex]; [latex]2[/latex]; [latex]2[/latex]
Davis: [latex]3[/latex]; [latex]3[/latex]; [latex]3[/latex]; [latex]4[/latex]; [latex]1[/latex]; [latex]4[/latex]; [latex]3[/latex]; [latex]2[/latex]; [latex]3[/latex]; [latex]1[/latex]
Maris: [latex]2[/latex]; [latex]3[/latex]; [latex]4[/latex]; [latex]4[/latex]; [latex]4[/latex]; [latex]6[/latex]; [latex]6[/latex]; [latex]6[/latex]; [latex]8[/latex]; [latex]3[/latex]
- Make a dot plot for the three authors and compare the shapes.
- Calculate the mean for each.
- Calculate the median for each.
- Describe any pattern you notice between the shape and the measures of center.
try it
1.
2.
The Ages Former U.S Presidents Died | |
---|---|
[latex]4[/latex] | [latex]6[/latex] [latex]9[/latex] |
[latex]5[/latex] | [latex]3[/latex] [latex]6[/latex] [latex]7[/latex] [latex]7[/latex] [latex]7[/latex] [latex]8[/latex] |
[latex]6[/latex] | [latex]0[/latex] [latex]0[/latex] [latex]3[/latex] [latex]3[/latex] [latex]4[/latex] [latex]4[/latex] [latex]5[/latex] [latex]6[/latex] [latex]7[/latex] [latex]7[/latex] [latex]7[/latex] [latex]8[/latex] |
[latex]7[/latex] | [latex]0[/latex] [latex]1[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex] [latex]4[/latex] [latex]7[/latex] [latex]8[/latex] [latex]8[/latex] [latex]9[/latex] |
[latex]8[/latex] | [latex]0[/latex] [latex]1[/latex] [latex]3[/latex] [latex]5[/latex] [latex]8[/latex] |
[latex]9[/latex] | [latex]0[/latex] [latex]0[/latex] [latex]3[/latex] [latex]3[/latex] |
Key: [latex]8|0 [/latex] means [latex]80[/latex]. |
3.
Concept Review
Looking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There are three types of distributions. A right (or positive) skewed distribution has a shape like Figure 3. A left (or negative) skewed distribution has a shape like Figure 2 . A symmetrical distribution looks like Figure 1.