§ 14.3 Numerical Summaries of Data

Numerical Summaries of a Data Set In the last section we looked at ways to graphically represent a data set--today we will look at numerical ways to summarize similar information. The are two major types of numerical summary: 1. Measures of location. 2. Measures of spread.

Numerical Summaries of a Data Set In the last section we looked at ways to graphically represent a data set--today we will look at numerical ways to summarize similar information. The are two major types of numerical summary: 1. Measures of location. 2. Measures of spread. average/mean range median interquartile range quartiles standard deviation

The Average / Mean The average or mean of a data set of size N is found by adding the numbers and dividing by N. Or more formally, if the data set is { x1 , x2 , x3 , . . . , xN } then the mean is given by: x1 + x2 + x3 + . . . + xN N

The Average / Mean What about when we are given a frequency table? Let’s look at the test scores from yesterday: 1 4 8 2 9 13 16 10 6 Frequency 96 76 72 64 60 56 48 44 40 36 32 28 24 Score

The Average / Mean From a Frequency Table Entering Data and Finding the Mean on the TI-83: Hit [Stat] Select “1: Edit…” Enter data into L1. If you are working from a frequency table enter the corresponding frequencies into L2. Go to the “List” menu ([2nd], [Stat]) Select “3: mean( “ You should now be on the ‘main’ screen. Proceed as follows: (a) If you are working from just a list of data, type “L1” ([2nd], [1]) , close the parentheses and hit [Enter]. (b) If you are working from a freq. table type “L1” followed by “,” and “L2” ([2nd], [2]). . Then close the parentheses and hit enter. Step 1: Calculate the total of the data. Total = x1 * f1 + x2 * f2+ x3* f3 + . . . + xk * f1k Step 2: Calculate N. N = f1 + f2 + f3 + . . . + fk Step 3: Calculate the average. Average = Total / N fk f2 . . . f1 Freque ncy xk x2 . . . x1 Data

Example: Average Salary The average salary at a local computer manufacturer with 50 employees is $42,000. The owner draws a yearly salary of $800,000. What is the average salary of the other 49 employees?

Example: 105 Exam Scores Suppose you have averaged a 132 out of 150 on the first 3 exams in Math 105. What score would you need on the fourth exam to have an average of 135?

Percentiles The p th percentile of a data set is the value such that p percent of the numbers fall at or below the value. The rest of the data falls at or above the value. We will call the p th percent of N the locator, and write it as L .

Example: Height Sorting Data on the TI-83: 1. Enter data into L1 as before. 2. Hit [Stat] 3. Select “2: SortA( “ 4. You should now be on the ‘main’ screen. Hit L1. ([2nd], 1) 5. Close the parentheses and hit enter.

Finding the p th Percentile Step 1: Sort the original data set by size. (Suppose {d1 , d2 , d3 , . . . , dN } is the sorted set) Step 2: Compute the value of the locator. L = ( p /100 )( N ) Step 3: The p th percentile is: (a) The average of dL and dL+1 if L is a whole number. (b) dL+ if L is not a whole number. L+ is L rounded up.

Percentiles: The Median and Quartiles The 50th percentile, called the median, is the percentile that is most commonly used. The median will be written M. The other two commonly used percentiles are the quartiles: The first quartile, written as Q1, is the 25th percentile. The third quartile, denoted Q3, is the 75th percentile.

Example: Let’s examine the test scores again. . . 1 4 8 2 9 13 16 10 6 Frequency 96 76 72 64 60 56 48 44 40 36 32 28 24 Score Find the quartiles and the median.

The Five-Number Summary One way to give a nice profile of a data set is the “five-number summary,” which consists of: 1. The lowest value, called the Min. 2. The first quartile, Q1. 3. The median, M. 4. The third quartile, Q3. 5. The highest value, called the Max.

Example: The Five-Number Summary for our test score example would look like this: 1 4 8 2 9 13 16 10 6 Frequency 96 76 72 64 60 56 48 44 40 36 32 28 24 Score

The Five-Number Summary: Box Plots We can also represent the Five- Number Summary graphically in what is called a box plot or a box-and-whiskers plot. Min M Q3 Max Q1

Example: Here is the box plot for our test score example: 1 4 8 2 9 13 16 10 6 Frequency 96 76 72 64 60 56 48 44 40 36 32 28 24 Score Min = 4 M = 44 Q3 = 48 Max = 96 Q1 = 36

§ 14.4 Measures of Spread

Example - Find the average and median of the following data sets:

The Range One way to measure the spread of data is to examine the range, given by R = Max - Min The problem with using the range is that outliers can severely affect it.

Example: Looking again at our ‘test score’ example. . . 1 4 8 2 9 13 16 10 6 Frequency 96 76 72 64 60 56 48 44 40 36 32 28 24 Score We see that the range with the outliers (4 and 96) would be R = 96 - 4 = 92. However, without those pieces of data we would have R = 76 - 24 = 52.

The Interquartile Range In order to eliminate the problems caused by outliers, we could make use of the interquartile range--the difference between the third and first quartile: IQR = Q3 - Q1 This measure tells us where the middle 50% of the data is located.

The IQR for this set of data is: IQR = Q3 - Q1 = 48 - 36 = 12 Example: Your instructor didn’t feel like making a different example. . . 1 4 8 2 9 13 16 10 6 Frequency 96 76 72 64 60 56 48 44 40 36 32 28 24 Score The IQR for this set of data is: IQR = Q3 - Q1 = 48 - 36 = 12

The Standard Deviation The idea: Measure how spread out your data set is by examining how far each piece of information is from some fixed reference point. The reference point we will use is the mean (average).

The Standard Deviation We could try to average the Deviations from the Mean: (Data value - Mean)

Example: Once again, the test score data. . . 1 4 8 2 9 13 16 10 6 Freque ncy 96 76 72 64 60 56 48 44 40 36 32 28 24 Score ( x ) 49.3 9 29.3 9 25.3 9 17.3 9 13.3 9 9.3 9 1.3 9 - 2.6 1 - 6.6 1 - 10.6 1 - 14.6 1 - 18.6 1 - 22.6 1 - 42.6 1 (x - 46.61)

The Standard Deviation We could try to average the Deviations from the Mean: (Data value - Mean) However, negative deviations and positive deviations will cancel each other out--in fact (assuming we don’t round off any of our figures) the average of the deviations from the mean will always be 0!

The Standard Deviation What would happen if we squared the deviations from the mean? The squared deviations are always non-negative, so there would be no canceling. The average of these squared deviations is called the variance, V.

The Standard Deviation Unfortunately, there is a problem with using the variance as well--the units of measure. For instance if we were studying people’s height in inches (in), the variance would appear in units of in2.

The Standard Deviation Unfortunately, there is a problem with using the variance as well--the units of measure. For instance if we were studying people’s height in inches (in), the variance would come be in units of in2. The solution to our dilemma is simple--we will just take the square root of the variance to get the what is called the standard deviation, .

Finding The Standard Deviation Step 1: Find the average/mean of the data set. Call it A. Step 2: For each number x in the data set find the deviation from the mean, x - A. Step 3: Square each of the deviations found in Step 2. Step 4: Find the average of the squared deviations found in step 3. This is the variance, V. Step 5: Take the square root of the variance. This is the standard deviation, .

Finding The Standard Deviation Another way to find the Standard Deviation by hand is to use the following formula: N ∑ ( xi - A )2 i=1 N √  =

Finding The Standard Deviation Finding all of the information from 14.2-14.3 on the TI-83: 1. Enter data as shown previously. Quit to the main screen. 2. Hit [Stat] 3. Move right to the “CALC“ menu. 4. Select “1-Var Stats”. 5. Now on the main screen, type “L1”. (If you are using data from a frequency table also type “,” and “L2”) Hit [Enter]. 6. Interpret the information as follows: x is the mean/average, A; x is the Standard Deviation; n is the size of your data set; If you arrow down the Min, Max, Median and quartiles should all be listed.

Example: Find the standard deviation for the following data set. {1, 6, 14, 19}