Numerical Summaries of Data Section 14.3

Transcription

1 MATH 11008: Numerical Summaries of Data Section 14.3 MEAN mean: The mean (or average) of a set of numbers is computed by determining the sum of all the numbers and dividing by the total number of observations. mean = x 1 + x x n n = xi n * The mean is affected by extremely high or extremely low scores. Example 1: Find the mean of each sample of numbers. (a) 83, 82, 81, 79 (b) 83, 82, 81, 21 Example 2: The class average on a math test was 36.5 points. The 18 girls in the class scored a total of 640 points. How many total points did the 9 boys score?

2 2 MATH 11008: NUMERICAL SUMMARIES OF DATA SECTION 14.3 MEDIAN Median: The median of a set of numbers is the value that lies in the middle of the data when arranged in ascending order. We will use M to represent the median. To find the median: Suppose x 1, x 2,, x n are the scores listed in increasing order. If n is odd, the median is the middle score in the list; namely, the n+1 2 score. If n is even, the median is the average of the two middle scores; namely, average the n 2 and n scores. * The median is not affected by extremely large or extremely small values. Example 3: Find the median of each sample of numbers. (a) 23, 24, 32, 35, 37, 45, 56, 57, 82 (b) 23, 25, 36, 39, 42, 45, 56, 57 MODE mode: of a variable is the most frequent observation of the variable that occurs in the data set. This is the only measure of central tendency that we can compute for nominal data. * There can be more than one mode; however, if each number appears equally often, then there is no mode. * Mode is not affected by other scores. Example 4: Give an example of a set of five data points that has two modes.

3 MATH 11008: NUMERICAL SUMMARIES OF DATA SECTION p-th percentile: The p-th percentile of a data set is the value such that p percent of the numbers fall at or below this value and rest fall at or above it. The p-th percentile splits the data into two parts: the lower p% of the data and the upper (100 p)% of the data. To find the p-th percentile of a data set: Suppose x 1, x 2,, x N are the scores listed in increasing order. ( p ) Find the locator L = N 100 Depending on whether L is a whole number or not, the p-th percentile is given by: the average of x L and x L+1 if L is a whole number. x L + where L + is L rounded up, if L is not a whole number. * The 25th percentile is called the lower quartile and is denoted Q 1. * the 75th percentile is called the upper quartile and is denoted Q 3. * CAUTION: There is no universally agreed upon procedure for computing percentiles, so different types of calculators and different statistical packages may give different answers. Example 5: Consider the following set of data points: 22, 24, 25, 27, 29, 35, 38, 45, 49, 53, 57, 62, 75, 76, 83 (a) Identify the score in the 80th percentile. (b) Identify the score in the 35th percentile.

4 4 MATH 11008: NUMERICAL SUMMARIES OF DATA SECTION 14.3 Example 6: The following table shows the ages of the firefighters in the local fire department. Age Frequency (a) Find the average age of the local firefighters. (b) Find the median age of the local firefighters. (c) Find the first quartile for this data set. (d) Find the third quartile for this data set. (e) Find the 30th percentile for this data set. (f) Find the 85th percentile for this data set.

5 MATH 11008: NUMERICAL SUMMARIES OF DATA SECTION Box Plot: A graphical representation of the median and other scores. To construct a box plot, we need to find 5 items, often referred to as the five number summary. 1. lowest score (minimum) 2. median 3. highest score (maximum) 4. lower quartile (Q 1 ) 5. upper quartile (Q 3 ) Interquartile Range (IQR): is the difference between the upper quartile and the lower quartile. That is, IQR = Q 3 Q 1. outlier: is any data point which lies more than 1.5 IQR units below lower quartile or more than 1.5 IQR units above upper quartile. Outliers are denoted by an asterisk. Drawing a modified box plot: 1. Determine the lower and upper fences: Lower fence = Q 1 1.5(IQR) Upper fence = Q (IQR) 2. Draw vertical lines at lower quartile Q 1, median, and upper quartile Q 3. Enclose these vertical lines in a box. 3. Locate the lower and upper fences. 4. Draw a horizontal line from the lower quartile Q 1 to the smallest data value that is larger than the lower fence. Draw a horizontal line from the upper quartile Q 3 to the largest data value that is smaller than the upper fence. 5. Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an asterisk ( ). * outlier lowest value not an outlier lower quartile median upper quartile highest value not an outlier * * outliers

6 6 MATH 11008: NUMERICAL SUMMARIES OF DATA SECTION 14.3 Example 7: The final medal standings for the top 15 countries at the 1996 Summer Olympics are listed below United States 101 Germany 65 China 63 Australia 50 France 41 Italy 37 South Korea 35 Cuba 27 Ukraine 25 Canada 23 Hungary 22 Romania 21 Netherlands 20 Poland 19 Spain 17 Determine the five number summary for this data, identify any outliers, and create a modified box plot for this data.