standard deviation of grouped data standard deviation can be found by summing the square of the...

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, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7

x f xf x - xbar (x – xbar)^2 (x – xbar) ^2 * f

5 26 37 68 29 1

Sum

Standard Deviation of Grouped Data• Find the sample standard deviation of the

following data:• 7, 6, 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7•

x f xf x - xbar (x – xbar)^2 (x – xbar) ^2 * f

5 2 106 3 187 6 428 2 169 1 9

Sum 14 95

786.614

95

n

xfx

Standard Deviation of Grouped Data• Find the sample standard deviation of the

following data:• 7, 6, 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7•

x f xf x - xbar (x – xbar)^2 (x – xbar) ^2 * f

5 2 10 -1.7866 3 18 -0.78577 6 42 0.21438 2 16 1.21439 1 9 2.2143

Sum 14 95

786.614

95

n

xfx

Standard Deviation of Grouped Data• Find the sample standard deviation of the

following data:• 7, 6, 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7•

x f xf x - xbar (x – xbar)^2 (x – xbar) ^2 * f

5 2 10 -1.786 3.18886 3 18 -0.7857 0.617357 6 42 0.2143 0.045928 2 16 1.2143 1.47459 1 9 2.2143 4.9031

Sum 14 95

786.614

95

n

xfx

Standard Deviation of Grouped Data• Find the sample standard deviation of the

following data:• 7, 6, 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7•

x f xf x - xbar (x – xbar)^2 (x – xbar) ^2 * f

5 2 10 -1.786 3.1888 6.3786 3 18 -0.7857 0.61735 1.8527 6 42 0.2143 0.04592 0.2768 2 16 1.2143 1.4745 2.9499 1 9 2.2143 4.9031 4.903

Sum 14 95 16.357

786.614

95

n

xfx

Standard Deviation of Grouped Data• Find the sample standard deviation of the

following data:• 7, 6, 7, 6, 7, 8, 5, 6, 7, 5, 7, 8, 9, 7•

x f xf x - xbar (x – xbar)^2 (x – xbar) ^2 * f

5 2 10 -1.786 3.1888 6.3786 3 18 -0.7857 0.61735 1.8527 6 42 0.2143 0.04592 0.2768 2 16 1.2143 1.4745 2.9499 1 9 2.2143 4.9031 4.903

Sum 14 95 16.357

786.614

95

n

xfx

Standard Deviation of Grouped Data

122.1

13

357.16

1

2

2

n

fxxss

•The calculator also gives 1.122

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

Sum 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

• The cell midpoints are 1, 2, 3, 4, and 5• The frequencies are 2, 3, 5, 4, and 1

1 2 3 4 50

1

2

3

4

5

6

Frequency versus Midpoints

Standard Deviation of Grouped Data

x f xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

1 22 33 54 45 1

Sum

• The cell midpoints are 1, 2, 3, 4, and 5• The frequencies are 2, 3, 5, 4, and 1

Standard Deviation of Grouped Data

x f xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

1 2 22 3 63 5 154 4 165 1 5

Sum 15 44

• The cell midpoints are 1, 2, 3, 4, and 5• The frequencies are 2, 3, 5, 4, and 1•

933.215

44

n

xfx

Standard Deviation of Grouped Data

x f xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

1 2 2 -1.9332 3 6 -0.9333 5 15 0.0674 4 16 1.0675 1 5 2.067

Sum 15 44

• The cell midpoints are 1, 2, 3, 4, and 5• The frequencies are 2, 3, 5, 4, and 1•

933.215

44

n

xfx What does this

add to?

Standard Deviation of Grouped Data

x f xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

1 2 2 -1.9332 3 6 -0.9333 5 15 0.0674 4 16 1.0675 1 5 2.067

Sum 15 44 0

• The cell midpoints are 1, 2, 3, 4, and 5• The frequencies are 2, 3, 5, 4, and 1•

933.215

44

n

xfx What does this

add to?

Standard Deviation of Grouped Data

x f xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

1 2 2 -1.933 -3.8672 3 6 -0.933 -2.8003 5 15 0.067 0.33334 4 16 1.067 4.26675 1 5 2.067 2.067

Sum 15 44 0

• The cell midpoints are 1, 2, 3, 4, and 5• The frequencies are 2, 3, 5, 4, and 1•

933.215

44

n

xfx What does this

add to?

Standard Deviation of Grouped Data

x f xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

1 2 2 -1.933 -3.8672 3 6 -0.933 -2.8003 5 15 0.067 0.33334 4 16 1.067 4.26675 1 5 2.067 2.067

Sum 15 44 0 0

• The cell midpoints are 1, 2, 3, 4, and 5• The frequencies are 2, 3, 5, 4, and 1•

933.215

44

n

xfx

Standard Deviation of Grouped Data

x f xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

1 2 2 -1.933 -3.867 3.738 7.4762 3 6 -0.933 -2.800 0.871 2.6133 5 15 0.067 0.3333 0.004 0.0224 4 16 1.067 4.2667 1.137 4.5515 1 5 2.067 2.067 4.271 4.271

Sum 15 44 0 0 18.933

• The cell midpoints are 1, 2, 3, 4, and 5• The frequencies are 2, 3, 5, 4, and 1•

•

933.215

44

n

xfx

Standard Deviation of Grouped Data

x f xf x - xbar (x – xbar) * f (x – xbar)^2 (x – xbar) ^2 * f

1 2 2 -1.933 -3.867 3.738 7.4762 3 6 -0.933 -2.800 0.871 2.6133 5 15 0.067 0.3333 0.004 0.0224 4 16 1.067 4.2667 1.137 4.5515 1 5 2.067 2.067 4.271 4.271

Sum 15 44 0 0 18.933

• The cell midpoints are 1, 2, 3, 4, and 5• The frequencies are 2, 3, 5, 4, and 1•

•

933.215

44

n

xfx

163.1

14

933.18

1

2

2

n

fxxss

• How can we do this in our calculator?• Put the “x” values in L1• Put the frequency in L2• Stat• Calc• 1-Var Stats• 2nd L1, 2nd L2, Enter

Standard Deviation of Grouped Data

Homework

• Pg 81 & 82, # 29 – 32 all (4)