How individual data points are positioned within a data set.

Transcription

1 Section 3.4 Measures of Position Percentiles How individual data points are positioned within a data set. P k is the value such that k% of a data set is less than or equal to P k. For example if we said that on a recent Math SAT exam, P 89 = 660, then 89% of the exam scores were less than or equal to 660. There are 99 percentiles, P 1 to P 99 which divide an ordered data set up into 100 roughly equal subsets. We need to be able to do two types of problems with percentiles. 1) Given a percentile, find the corresponding data value. Consider the following data set representing the ages of actresses when they won the Academy Award. There are 76 data points in this data set Note that this data set is all ready ordered, if a data set is not ordered you MUST order the data first. This can be done with the calculator. Find P 37, the data value of the 37 th percentile. a. Step 1: Find the position of the desired point. Let i (for index) be the position. k i n 100 b. If i is not a whole number round UP to the next whole number. Note this is a round up not a round off!

2 c. Count through the list to the value in the ith position. This is the value of the desired percentile. Interpretation: d. Find the value of the 75 th percentile. We will have to do something slightly different here. Remember: If i is not a whole number, round up to the next whole number. If i is a whole number the percentile value will be the average of the ith element and the NEXT elemenet. Most of the time we will be interested in finding 3 important percentiles, called the quartiles. These are: Example: Suppose the following data set represents the amount of rain in a city in 12 randomly chosen months. All value are measured in inches Find the quartiles for this data set using the calculator. Interpret each value.

3 Z-scores Another important measure of position is the z-score. The z-score is the number of standard deviations that a data point is positioned above or below the mean. Z-scores are always rounded to 2 decimal places. The z-score is positive if a value is above the mean, negative if the value is below the mean. Formula: data value mean z std. dev. Example: Suppose you take a math exam and score 88. The mean score on the exam was 74 and the standard deviation was 7.5. Find and interpret the z-score. Example: Suppose that you took and English exam and scored 69, the mean was 45 and the standard deviation was Find and interpret the z-score. z-scores can be sued to compare values, even though they come from different populations. Relatively speaking, which score was higher, math or English? Example: With a height of 75 inches, Lyndon Johnson was the tallest president of the last century. With a height of 85 inches, Shaquille O Neill was the tallest basketball player on his Miami team. Who is relatively taller? Presidents of the last century have heights with a mean of 71.5 in. and a standard

4 deviation of 2.1 in. The Miami basketball team had heights with a mean of 80 in. and a standard deviation of 3.3 in. Z-scores can be used to identify usual, unusual and outlier values. 1. A data point is considered a usual value if 2. A data point is considered an unusual value if 3. A data point is considered an outlier if Def: An outlier is a data point that is more than standard deviations away from the mean. Suppose that you had scored 98 on the math exam, instead of 88. Find the z-score and comment on this value. Section 3.5 Chebyshev s Rule and the Emperical Rule Chebyshev s Rule- The proportion of data values from a data set that fall within k standard deviations fo the mean will be at least: % k 1 2 k

5 Example: For the math exam where the mean was 74 and the standard deviation 7.5, what percentage fo the data was within 2 standard deviations of the mean. Find the corresponding interval and interpret your answer. Suppose that a data set has a mean of 604 and a standard deviation of 112. What does Chebyshev s Rule say about the percentage of the data between 436 and 772? Emperical Rule Another rule that is helpful in interpreting value for a standard deviation is the Emperical Rule. The Emperical Rule applies ONLY to bell shaped data sets. Emperical Rule In an approximately bell-shaped data set, the following properties apply. 1) About 68% of all data values fall within one standard deviation of the mean 2) About 95% of all data values fall within two standard deviations of the mean 3) About 99.7% of all data values fall within three standard deviations of the mean. What does this mean for intervals and z-scores?

6 Draw the curves that go with the Emperical Rule: Example: The mean amount spent per person on holiday gifts is $608 with a standard deviation of $174. The data is approximately bell shaped. a. What can be said about the percentage of this data set with in $86 and $1130? b. Between $434 and $956? c. Above $1130?

7 Section 3.6 Robust Measures A robust measure is a value that is not sensitive to outliers. The mean and standard deviation are sensitive to outliers and therefore not robust. The median and the quartiles and the percentiles are not sensitive to outliers and is therefore considered robust. In this section we look at other robust measures. Example: Suppose 8 students take the SAT Math exam and their scores are as follows: Use your calculator to find the 5-number summary for this data set. The 5-number summary is: 1) Minimum value 2) Q 1 3) Med 4) Q 3 5) Maximum Value Definition: The interquartile range or the IQR = Q 3 -Q 1. The IQR is the spread of the middle 50% of the data set. Q 1 is called the lower hinge of the data set. Q 3 is called the upper hinge of the data set. Find and interpret the interquartile range for the above data set. The quartiles are also used to determine outliers. Define: 1) Lower Fence = Q 1 1.5IQR 2) Upper Fence = Q IQR

8 Find the upper and lower fences for the above data set. A data point is considered an outlier if it is less than the lower fence or greater than the upper fence. Are there any outliers in this data set? Example: p. 156 in your text. Looking at table 3.29 of the median household incomes for 16 states. The 5-number summary for this data set is 1) ) ) ) ) 82.4 (These can be found on your calculator) From this find each of the following: Lower hinge Upper hinge IQR Lower Fence Upper Fence Are there any outliers in the data set?

9 One more graph: The Boxplot Draw a boxplot for the above data set. 1) Find the upper and lower fences for the data set. 2) Draw a horizontal number line for the data set. Above the number line plot the quartiles and the median. Connect the lines for Q1 and Q3 to form a box. 3) Temporarily indicate the fences as brackets, [ and ] above the number line 4) Draw a horizontal line from Q1 to the smallest data value greater than the lower fence. Do the same for Q3 to the largest data value lower than the upper fence. (You are omitting the outliers at this point.) 5) Indicate any values above or below the fences with a *