mean, median, mode, and midrange of grouped data section 2.5
TRANSCRIPT
Mean, Median, Mode, and Midrange of Grouped Data
Section 2.5
Grouped Data……
You must add one more column than you did using ungrouped data.
You now need a midpoint column.
The symbol for the midpoint is . mx
Formulas……Mean
Mean
n
xfx m
Median
There IS a formula to find the median using grouped data.
mLwf
cfn
median
)(2
Mode……
Find the greatest frequency and read across the chart until you see the class that corresponds to it.
Your answer will be the entire interval.
Midrange…..
Add the lowest number in the first row to the highest number in the last row.
Divide that answer by 2.
Example…..
Find the mean, median, and mode of the set of grouped data.
x f
6-11 1
11-16 2
16-21 3
21-26 5
26-31 4
31-36 3
36-41 2
n=20
Here is the list you should have……
x f midpoint f x midpoint cf
6-11 1 8.5 8.5 1
11-16 2 13.5 27 3
16-21 3 18.5 55.5 6
21-26 5 23.5 117.5 11
26-31 4 28.5 114 15
31-36 3 33.5 100.5 18
36-41 2 38.5 77 20
20 500
Mean……
2520
500x
Median…..
n/2 = 20/2 = 10x f midpoint f(midpoint) cf
6-11 1 8.5 8.5 1
11-16 2 13.5 27 3
16-21 3 18.5 55.5 6
21-26 5 23.5 117.5 11
26-31 4 28.5 114 15
31-36 3 33.5 100.5 18
36-41 2 38.5 77 20
20 500
Plug values into formula….
2521)5(5
610
median
Mode and Midrange……
The mode is 21-26.
Midrange =
x f midpoint f(midpoint) cf
6-11 1 8.5 8.5 1
11-16 2 13.5 27 3
16-21 3 18.5 55.5 6
21-26 5 23.5 117.5 11
26-31 4 28.5 114 15
31-36 3 33.5 100.5 18
36-41 2 38.5 77 20
20 500
5.232
47
2
)641(
Now you try……….
Find the mean, median, and mode of the following set.
x f
63-66 2
66-69 4
69-72 8
72-75 5
75-78 2
n=21
Your finished list…….
x f midpoint f(midpoint) cf
63-66 2 64.5 129 2
66-69 4 67.5 270 6
69-72 8 70.5 564 14
72-75 5 73.5 367.5 19
75-78 2 76.5 153 21
n=21 1483.5
Mean……
6.7021
5.1483x
Median…….
7.7069)3(8
65.10
median
Mode……..
The mode is 69-72.x f midpoint f(midpoint) cf
63-66 2 64.5 129 2
66-69 4 67.5 270 6
69-72 8 70.5 564 14
72-75 5 73.5 367.5 19
75-78 2 76.5 153 21
n=21 1483.5
Midrange……..
The midrange =
(63 + 78)/2 = 70.5 x f midpoint f(midpoint) cf
63-66 2 64.5 129 2
66-69 4 67.5 270 6
69-72 8 70.5 564 14
72-75 5 73.5 367.5 19
75-78 2 76.5 153 21
n=21 1483.5
Range, Variance and St. Deviation – GroupedSection 2.5
Grouped Data……
Variance Formula
1
/222
n
nxfxfs mm
Standard Deviation
2ss
Range…..
High number in last row minus low number in first row.
Example……
Find the variance, standard deviation, and range of the set.
x f
2-8 12
8-14 4
14-20 6
20-26 22
26-32 8
n=52
Calculator Steps……
Put lower boundaries in L1 and upper boundaries in L2. Put frequencies in L3. Set a formula for midpoint in L4.
Find f times midpoint by setting a formula in L5.
Find f times midpoint squared in L6 by setting a formula.
Your lists should look like this……
x f midpoint f (midpoint) f ( midpoint sq)
2-8 12 5 60 300
8-14 4 11 44 484
14-20 6 17 102 1734
20-26 22 23 506 11638
26-32 8 29 232 6728
n=52 944 20884
Find the variance.
1
/222
n
nxfxfs mm
5.7351
52
94420884
2
2
s
Find the standard deviation.
2ss
6.846606335.73 s
Range = High - Low
Range = 32 – 2 = 30
Example……
Find the mean, median, mode, midrange, range, variance, and st. deviation of the data set.
x f
10 - 15 5
15 - 20 9
20 - 25 7
25 - 30 3
30 - 35 2
Here are the lists……
x f midpoint f x midpoint f x mp squared cf
10 - 15 5 12.5 62.5 781.25 5
15 - 20 9 17.5 157.5 2756.3 14
20 - 25 7 22.5 157.5 3543.8 21
25 - 30 3 27.5 82.5 2268.8 24
30 - 35 2 32.5 65 2112.5 26
n=26 525 11462.5
Mean:
2.2026
525
x
n
xfx m
Median……
4.19
15)5(9
)513(
15)5(9
5226
)(2
med
med
med
Lwf
cfnmed m
Mode……
Greatest Frequency is 9.
Mode = 15-20
Midrange……
5.222
452
)1035(2
)(
midrange
midrange
LowHighmidrange
Range……
25
1035
Range
Range
LowHighRange
Variance……
5.3425
265255.11462
1
)(
2
2
22
2
s
nn
xfxfs
mm
St. Deviation……
9.546153846.34
2
s
ss
Homework…….
Find the measures of center and variation for the grouped data on HW3.