Position of a Term in a Distribution for AP Statistics
Practice problems for these concepts can be found at:
- One-Variable Data Analysis Multiple Choice Practice Problems for AP Statistics
- One-Variable Data Analysis Free Response Practice Problems for AP Statistics
- One-Variable Data Analysis Review Problems for AP Statistics
- One-Variable Data Analysis Rapid Review for AP Statistics
Up until now, we have concentrated on the nature of a distribution as a whole. We have been concerned with the shape, center, and spread of the entire distribution. Now we look briefly at individual terms in the distribution.
There are positions in a dataset that give us valuable information about the dataset. The five-number summary of a dataset is composed of the minimum value, the lower quartile, the median, the upper quartile, and the maximum value. On the TI-83/84, these are reported on the second screen of data when you do 1-Var Stats as: minX, Q1, Med, Q3, and maxX.
example: The following data are standard of living indices for 20 cities: 2.8, 3.9, 4.6, 5.3, 10.2, 9.8, 7.7, 13, 2.1, 0.3, 9.8, 5.3, 9.8, 2.7, 3.9, 7.7, 7.6, 10.1, 8.4, 8.3. Find the 5-number summary for the data.
solution: Put the 20 values into a list on your calculator and do 1-Var Stats. We find: minX=0.3, Q1=3.9, Med=7.65, Q3=9.8, and maxX=13
Boxplots (Outliers Revisited)
In additon to the three types of graphs: dotplot, stemplot, and histogram. Using the five-number summary, we can add a fourth type of one-variable graph to this group: the boxplot. A boxplot is simply a graphical version of the five-number summary. A box is drawn that contains the middle 50% of the data (from Q1 to Q3) and "whiskers" extend from the lines at the ends of the box (the lower and upper quartiles) to the minimum and maximum values of the data if there are no outliers. If there are outliers, the "whiskers" extend to the last value before the outlier that is not an outlier.
The outliers themselves are marked with a special symbol, such as a point, a box, or a plus sign.
The boxplot is sometimes referred to as a box and whisker plot.
example: Consider again the data from the previous example: 2.8, 3.9, 4.6, 5.3, 10.2, 9.8, 7.7, 13, 2.1, 0.3, 9.8, 5.3, 9.8, 2.7, 3.9, 7.7, 7.6, 10.1, 8.4, 8.3. A boxplot of this data, done on the TI-83/84, looks like this (the five-number summary was [0.3, 3.9, 7.65, 9.8, 13]):
example: Using the same dataset as the previous example, but replacing the 10.2 with 20, which would be an outlier in this dataset (the largest possible non-outlier for these data would be 9.8 + 1.5(9.8 – 3.9) = 18.65), we get the following graph on the calculator:
Note that the "whisker" ends at the largest value in the dataset, 13, which is not an outlier.
Percentile Rank of a Term
The percentile rank of a term in a distribution equals the proportion of terms in the distribution less than the term. A term that is at the 75th percentile is larger than 75% of the terms in a distribution. If we know the five-number summary for a set of data, then Q1 is at the 25th percentile, the median is at the 50th percentile, and Q3 is at the 75th percentile. Some texts define the percentile rank of a term to be the proportion of terms less than or equal to the term. By this definition, being at the 100th percentile is possible.
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