Measures of Dispersion Week 3

What is dispersion? Dispersion is how the data is spread out, or dispersed from the mean. The smaller the dispersion values, the more consistent the data. The larger the dispersion values, the more spread out the data values are. This means that the data is not as consistent.

Consider these sets of data: Grades from Test # 1 = - 81,83,83,82,86,81,87,80,81,86 Grades from Test # 2 = - 95,74,65,90,87,97,60,81,99,76 What differences do you see between the two sets? What are the Mean scores? Ranges? Do you believe these grades tell a story?

Important Symbols to remember = mean X = an individual value N = Population size n = sample population size i = 1 st data value in population

The average of the squares of each difference of a data value and the mean. Variance

is the measure of the average distance between individual data points and their mean. It is the square root of the variance. The lower case Greek letter sigma is used to denote standard deviation. Standard Deviation

How to Calculate Standard Deviation Given the data set {5, 6, 8, 9}, calculate the standard deviation. Step 1: find the mean of the data set

How to Calculate Standard Deviation Step 2: Find the difference between each data point and the mean.

Step 3: Square the difference between each data point and the mean. How to Calculate Standard Deviation

Step 4: Sum the squares of the differences between each data point and the mean. How to Calculate Standard Deviation

Step 5: Take the square root of the sum of the squares of the differences divided by the total number of data points; * The average distance between individual data points and the mean is units from 7

Standard Deviation Formula of what we just did: For sample S.D. use 1/(n-1)

When to use Pop. vs. Sample When we have the actual entire population (for example our class, 29 students), we would use the Population formula. If the problem tells us to use a particular formula; Pop. v. Samp. If we are working with less entire population of a much larger group, we will use the sample formula. (Which is one taken away from the pop. total)

Why is this useful? It provides clues as to how representative the mean is of the individual data points. For example, consider the following two data sets with the same means, but different standard deviations. Bowler # 1 {98, 99, 101, 102} Bowler # 2 {30,51, 149, 169} The mean with the standard deviation provides a better description of the data set.

TI-83 to Calculate Standard Deviation. Step 1. Press STAT,EDIT,1:EDIT Step 2. Enter your data in the L1 column, pressing enter after every data entry. Step 3. Press STAT, CALC,1-Var stats Step 4. Scroll down to the lower case symbol for the Greek letter sigma calculator help.

Let’s try one more by hand: (1)Find the population standard deviation for the following Stats class test grades: 78, 84, 88, 92, 68, 82, 92, 72, 88, 86, 76, 90 (a) How many grades fall within one SD of the mean? (b) What percent fall within one SD of the mean? * Now check it with the calculator!