Standard Deviation of Grouped Data Standard deviation can

Standard Deviation of Grouped Data • Standard deviation can be found by summing the square of the deviation of each value, or, • If the value is present more than once, the square of the deviation can be calculated once and multiplied by the frequency of occurrences

Standard Deviation of Grouped Data • Find the sample standard deviation of the following data: • 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7 Sum x f 5 2 6 3 7 6 8 2 9 1 xf x - xbar (x – xbar)^2 (x – xbar) ^2 * f

Standard Deviation of Grouped Data • Find the sample standard deviation of the following data: • 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7 • Sum x f xf 5 2 10 6 3 18 7 6 42 8 2 16 9 14 95 x - xbar (x – xbar)^2 (x – xbar) ^2 * f

Standard Deviation of Grouped Data • Find the sample standard deviation of the following data: • 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7 • Sum x f xf x - xbar 5 2 10 -1. 786 6 3 18 -0. 7857 7 6 42 0. 2143 8 2 16 1. 2143 9 1 9 2. 2143 14 95 (x – xbar)^2 (x – xbar) ^2 * f

Standard Deviation of Grouped Data • Find the sample standard deviation of the following data: • 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7 • Sum x f xf x - xbar (x – xbar)^2 5 2 10 -1. 786 3. 1888 6 3 18 -0. 7857 0. 61735 7 6 42 0. 2143 0. 04592 8 2 16 1. 2143 1. 4745 9 1 9 2. 2143 4. 9031 14 95 (x – xbar) ^2 * f

Standard Deviation of Grouped Data • Find the sample standard deviation of the following data: • 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7 • Sum x f xf x - xbar (x – xbar)^2 (x – xbar) ^2 * f 5 2 10 -1. 786 3. 1888 6. 378 6 3 18 -0. 7857 0. 61735 1. 852 7 6 42 0. 2143 0. 04592 0. 276 8 2 16 1. 2143 1. 4745 2. 949 9 1 9 2. 2143 4. 9031 4. 903 14 95 16. 357

Standard Deviation of Grouped Data • Find the sample standard deviation of the following data: • 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7 • Sum x f xf x - xbar (x – xbar)^2 (x – xbar) ^2 * f 5 2 10 -1. 786 3. 1888 6. 378 6 3 18 -0. 7857 0. 61735 1. 852 7 6 42 0. 2143 0. 04592 0. 276 8 2 16 1. 2143 1. 4745 2. 949 9 1 9 2. 2143 4. 9031 4. 903 14 95 16. 357

Standard Deviation of Grouped Data • The calculator also gives 1. 122 Sum x f xf x - xbar (x – xbar)^2 (x – xbar) ^2 * f 5 2 10 -1. 786 3. 1888 6. 378 6 3 18 -0. 7857 0. 61735 1. 852 7 6 42 0. 2143 0. 04592 0. 276 8 2 16 1. 2143 1. 4745 2. 949 9 1 9 2. 2143 4. 9031 4. 903 14 95 16. 357

Standard Deviation of Grouped Data • We grouped the data in the above example. • The same process can be used when given data in the form of a histogram or pie chart. • Since the values of the specific data points has been lost, assume all the data points within a cell have the same value as the cell midpoint. • The student is left to review Example 10 on page 77.

Standard Deviation of Grouped Data • Assume the histogram on the following slide represents our data. • Make a table of values (x values – the midpoint of each column), including the frequency of each column. • Calculate the sample standard deviation of the data represented in the histogram

Standard Deviation of Grouped Data Frequency versus Midpoints 6 5 4 3 2 1 0 1 2 3 4 5 • The cell midpoints are 1, 2, 3, 4, and 5 • The frequencies are 2, 3, 5, 4, and 1

Standard Deviation of Grouped Data • The cell midpoints are 1, 2, 3, 4, and 5 • The frequencies are 2, 3, 5, 4, and 1 Sum x f 1 2 2 3 3 5 4 4 5 1 xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

Standard Deviation of Grouped Data • The cell midpoints are 1, 2, 3, 4, and 5 • The frequencies are 2, 3, 5, 4, and 1 • Sum x f xf 1 2 2 2 3 6 3 5 15 4 4 16 5 15 44 x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

Standard Deviation of Grouped Data • The cell midpoints are 1, 2, 3, 4, and 5 • The frequencies are 2, 3, 5, 4, and 1 • What does this add to? Sum x f xf x - xbar 1 2 2 -1. 933 2 3 6 -0. 933 3 5 15 0. 067 4 4 16 1. 067 5 1 5 2. 067 15 44 (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

Standard Deviation of Grouped Data • The cell midpoints are 1, 2, 3, 4, and 5 • The frequencies are 2, 3, 5, 4, and 1 • What does this add to? Sum x f xf x - xbar 1 2 2 -1. 933 2 3 6 -0. 933 3 5 15 0. 067 4 4 16 1. 067 5 1 5 2. 067 15 44 0 (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

Standard Deviation of Grouped Data • The cell midpoints are 1, 2, 3, 4, and 5 • The frequencies are 2, 3, 5, 4, and 1 • What does this add to? Sum x f xf x - xbar (x – xbar) * f 1 2 2 -1. 933 -3. 867 2 3 6 -0. 933 -2. 800 3 5 15 0. 067 0. 3333 4 4 16 1. 067 4. 2667 5 1 5 2. 067 15 44 0 (x – xbar)^2 (x – xbar) ^2 * f

Standard Deviation of Grouped Data • The cell midpoints are 1, 2, 3, 4, and 5 • The frequencies are 2, 3, 5, 4, and 1 • Sum x f xf x - xbar (x – xbar) * f 1 2 2 -1. 933 -3. 867 2 3 6 -0. 933 -2. 800 3 5 15 0. 067 0. 3333 4 4 16 1. 067 4. 2667 5 1 5 2. 067 15 44 0 0 (x – xbar)^2 (x – xbar) ^2 * f

Standard Deviation of Grouped Data • The cell midpoints are 1, 2, 3, 4, and 5 • The frequencies are 2, 3, 5, 4, and 1 • • Sum x f xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f 1 2 2 -1. 933 -3. 867 3. 738 7. 476 2 3 6 -0. 933 -2. 800 0. 871 2. 613 3 5 15 0. 067 0. 3333 0. 004 0. 022 4 4 16 1. 067 4. 2667 1. 137 4. 551 5 2. 067 4. 271 15 44 0 0 18. 933

Standard Deviation of Grouped Data • The cell midpoints are 1, 2, 3, 4, and 5 • The frequencies are 2, 3, 5, 4, and 1 • • Sum x f xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f 1 2 2 -1. 933 -3. 867 3. 738 7. 476 2 3 6 -0. 933 -2. 800 0. 871 2. 613 3 5 15 0. 067 0. 3333 0. 004 0. 022 4 4 16 1. 067 4. 2667 1. 137 4. 551 5 2. 067 4. 271 15 44 0 0 18. 933

Standard Deviation of Grouped Data • • How can we do this in our calculator? Put the “x” values in L 1 Put the frequency in L 2 Stat Calc 1 -Var Stats 2 nd L 1, 2 nd L 2, Enter

Homework • Pg 81 & 82, # 29 – 32 all (4)