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