Statistics2.5: Measures of Position
Objective 1: I can find the quartiles of a data set, the interquartile range, and represent a data set using a box-and-whisker plot.
In this section we will learn to use ______to specify the position of data in a data set.
______are numbers that divide a data set into ______parts. Here are some common fractiles that you may have heard of before.
Median:
Quartile:
Percentile:
Decile:
The three quartiles, ____, ____ (or the ______) and ____, divide a data set into ____ equal parts. In this objective we are going to focus on finding the ______, which consists of the values:
Each section between two values contains ______of the data values.
Read Example 1, pg 102
TIY 1: Find the 5-number summary for the data set below.
1938227182429152231
2725161112319202814
Using technology to find the 5-number summary:
TIY 2: The tuition costs (in thousands of dollars) for 25 universities are listed. Use a calculator or computer to find the 5-number summary.
20262825311423151226293431
193117151720313216212228
After finding the quartiles, you can then find the ______or ______.
IQR =
Read Example 3, pg 104
TIY 3: Find the IQR for the data in TIY 2.
The IQR is a measure of variation that gives you an idea of how much the middle 50% of the data varies. It can also be used to identify ______. An outlier is any data value that lies outside the interval found by
*Are there any outliers in our data set from TIY 2?
One application of quartiles is to represent data with a ______plot. To draw a box-and-whisker plot,
1)
2)
3)
4)
5)
Read Example 4, pg 105
TIY 4: Create a box-and-whisker plot of the data in TIY 2.
Objective 2: I can interpret percentiles and other fractiles.
Besides using the median and quartiles to partition data, we can use other fractiles. They even have their own notation.
Median
Deciles
Quartiles
Percentiles
A fractile is used to compare data values in the same set. For example, if a child places in the first quartile on a standardized test, then that child performed better than ______of the other children. When you receive your ACT scores back, they grade you in each test category by ______. So if you place in the 88th percentile for ACT math, then you performed better than ______of other students who took that math test.
Read “Insight” and “Study Tip” on page 106.
Read Example 5, pg 106.
TIY 5: The number of touchdowns scored by all Division a! football teams are represented by the ogive below. Use it to answer parts (a) and (b).
a) Find the percentile that corresponds to 40 touchdowns. Interpret this.
b) Find the number of touchdowns for a team that scored in the 75th percentile. Interpret this.
Objective 3: The standard score
When you know the mean and standard deviation of a data set, you can measure a data value’s position in the data set with a ______or ______. The ______, or z-score, represents the number of ______a given value, x, falls from the ______. The z-score for any x-value in a data set can be found by the formula:
A ______z-score means the data value is ______the mean.
A ______z-score means the data value is ______the mean.
A z-score of ______means the data value _____ the mean.
Z-scores are often used to compare data from different sets—such as ACT scores compared to SAT scores. How can you tell which score is “better” for its data set?
Quick Example: Bob scored 27 on the ACT which has a mean of 21 and a SD of 3.5. He also took the SAT math and scored 680, which has a mean of 500 and a SD of 100. On which test did Bob perform better? (Always round z-score to 2 decimal places.)
Z-scores can also help us determine if a data value is an ______. Recall that an outlier is any data value that is ______.
Read Example 6, pg 107
TIY 6: The monthly utility bills in a city have a mean of $70 and a standard deviation of $8. Find the z-scores that correspond to utility bills of $60, $71, and $92. Would any of the bills be considered outliers?
Read Example 7, pg 108
TIY 7: Alan Arkin won a Best Supporting Actor Oscar at age 72, while Jennifer Hudson won a Best Supporting Actress Oscar at age 25. The mean age of best supporting actor is 50.1 with a SD of 13.9. The mean age for best supporting actress is 39.7 with a SD of 14. Find the z-score that corresponds to the age for each person and compare your results.