Download f (midpoint)

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
xm .
Formulas……Mean

Mean
 f  xm
x
n
Median

There IS a formula to find the median
using grouped data.
n
 cf
2
median 
( w)  Lm
f
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
6-11
11-16
16-21
f
1
2
3
21-26
26-31
31-36
5
4
3
36-41
2
n=20
Here is the list you should
have……
x
f
midpoint
f x midpoint
cf
6-11
11-16
16-21
1
2
3
8.5
13.5
18.5
8.5
27
55.5
1
3
6
21-26
26-31
31-36
5
4
3
23.5
28.5
33.5
117.5
114
100.5
11
15
18
36-41
2
38.5
77
20
20
500
Mean……
500
x
 25
20
Median…..

n/2 = 20/2 = 10
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
Plug values into formula….
10  6
median 
(5)  21  25
5
Mode and Midrange……


The mode is 21-26.
Midrange =
(41  6) 47

 23.5
2
2
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
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
66-69
69-72
72-75
75-78
2
4
8
5
2
64.5
67.5
70.5
73.5
76.5
129
270
564
367.5
153
2
6
14
19
21
n=21
1483.5
Mean……
1483.5
x
 70.6
21
Median…….
10.5  6
median 
(3)  69  70.7
8
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 –
Grouped
Section 2.5
Grouped Data……

Variance Formula
 f  xm   f  xm  / n
s 
n 1
2
2
2

Standard Deviation
s
s
2
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
944
20884
n=52
 f  xm   f  xm  / n
s 
n 1
2
2
2

Find the variance.
2
944
20884 
2
52
s 
 73.5
51
s

s
2
Find the standard deviation.
s  73.46606335  8.6
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
525
11462.5
n=26
Mean:
 f  xm
x
n
525
x
 20.2
26
Median……
med
med
med
med
n  cf
 2
( w)  Lm
f
26  5
 2
(5)  15
9
(13  5)

(5)  15
9
 19.4
Mode……

Greatest Frequency is 9.

Mode = 15-20
Midrange……
( High  Low)
midrange 
2
(35  10)
midrange 
2
45
midrange 
 22.5
2
Range……
Range  High  Low
Range  35  10
Range  25
Variance……
s 
2
s 
2
 f x 
2
m
( f  x m )
n 1
2
11462.5  525
25
2
n
26  34.5
St. Deviation……
s s
2
s  34.46153846  5.9
Homework…….

Find the measures of center and variation
for the grouped data on HW3.