Mean from a Frequency Table
Calculating the Mean: If there are large amounts of data, it is
easier if it is displayed in a frequency table.
Example 1.
The number of goals scored by a team in 20 games are given below :
3,2,4,2,2,3,2,2,0,5,1,1,2,3,0,2,1,4,1,0
Goals x
Frequency, f
f.x
0
3
0
1
4
4
2
7
14
3
3
9
4
2
8
5
1
5
Mode
∑f= 20
∑fx= 40
Mean = ∑fx
∑f
= 40
20
=2
Median from a Frequency Table
Calculating the median If there are large amounts of data, it is
easier if it is displayed in a frequency table.
Example 1.
The number of goals scored by a team in 20 games are given below :
3,2,4,2,2,3,2,2,0,5,1,1,2,3,0,2,1,4,1,0
(20)/2
Goals x
Frequency, f
C. F.
0
3
3
= 10
1
4
7
2
7
14
(20)/2 + 1
= 11
3
3
17
4
2
19
5
1
20
Mode
∑f= 20
The 10th value is 2
The 11th value is 2
∴ MEDIAN = ( 2+2 ) / 2 = 4/2 = 2
Grouped Data
Large quantities of data can be much more easily viewed and managed if
placed in groups in a frequency table. Grouped data does not enable
exact values for the mean, median and mode to be calculated. Alternate
methods of analyising the data have to be employed.
Example 1.
During 3 hours at Heathrow airport 55 aircraft arrived late. The number of
minutes they were late is shown in the grouped frequency table below.
Data is
grouped
into 6 class
intervals of
width 10.
minutes late
frequency
0-9
27
10 - 19
10
20 - 29
7
30 - 39
5
40 - 49
4
50 - 59
2
Grouped Data
Estimating the Mean: An estimate for the mean can be obtained by
assuming that each of the raw data values takes the midpoint value of
the interval in which it has been placed.
Example 1.
During 3 hours at Heathrow airport 55 aircraft arrived late. The number of
minutes they were late is shown in the grouped frequency table below.
minutes Late
Frequency,f midpoint(c.c.) F × c.c.
0-9
27
4.5
10 - 19
10
14.5
20 - 29
7
30 - 39
5
24.5
34.5
40 - 49
4
44.5
178
54.5
109
50 - 59
2
f  55
121.5
145
171.5
172.5
 f  c.c.  897.5
Mean estimate = 897.5/55 ≈ 16.32 minutes
Grouped Data
The Modal Class
The modal class is simply the class interval of highest frequency.
Example 1.
During 3 hours at Heathrow airport 55 aircraft arrived late. The number of
minutes they were late is shown in the grouped frequency table below.
minutes late
frequency
0-9
27
10 - 19
10
20 - 29
7
30 - 39
5
40 - 49
4
50 - 59
2
Modal class = 0 - 9
‫‪worksheet‬‬
‫بالنسبة لمجموعة البيانات التالية أوجد ‪:‬‬
‫‪For the following set of data find :‬‬
‫‪6 , 8 , 5 , 11 , 3 , 1 , 7 , 9 , 3‬‬
‫‪ )1‬المتوسط الحسابي‬
‫‪1) The mean‬‬
‫‪ )2‬الوسيط‬
‫‪2) The median‬‬
‫‪ )3‬المنوال‬
‫‪3) The mode‬‬
‫‪ )4‬المدى‬
‫‪4) The range‬‬
‫بالنسبة لمجموعة البيانات التالية أوجد ‪:‬‬
‫‪ )1‬المتوسط الحسابي‬
‫‪For the following set of data find :‬‬
‫‪9 , 3 , 8 , 7 , 1 , 9 , 11 , 4 , 3 , 2‬‬
‫‪1) The mean‬‬
‫‪ )2‬الوسيط‬
‫‪2) The median‬‬
‫‪ )3‬المنوال‬
‫‪3) The mode‬‬
‫‪ )4‬المدى‬
‫‪4) The range‬‬
worksheet
The ages of a random sample of 30 persons are given in the table :
: ‫ شخص كما بالجدول‬30 ‫أعمار عينة عشوائية من‬
Age ( x )
(f)
40
2
41
7
42
9
43
6
44
5
45
1
total
X.f
(cf)
Find :
1) The mean age
2) The median
3) The mode
4) The range of the ages
: ‫أوجد‬
‫) المتوسط الحسابي لألعمار‬1
‫) الوسيط لألعمار‬2
‫) المنوال‬3
‫) مدى األعمار‬4
worksheet
The following frequency distribution represents the lengths of 20 persons
‫ شخص‬20 ‫التوزيع التكراري يمثل أطوال‬
intervals
(f)
150 - 154
2
155 - 159
8
160 - 164
5
165 - 169
4
170 - 179
1
c.c.
c.c.× f
Find :
: ‫أوجد‬
1) The mean of the lengths
‫) المتوسط الحسابي لألطوال‬1
total
2) The model class and estimate the mode
3) The range of the lengths
‫) اكتب الفئة المنوالية وقدر المنوال‬2
‫) أوجد مدى األطوال‬3
worksheet
The grades of 25 students are given below :
: ‫ طالب كما يلي‬25 ‫درجات‬
42 , 63 , 47 , 77 , 46 , 71 , 68 , 83 , 91 , 55 , 67 , 66 , 63 , 57 , 50 , 69 , 73 ,
82, 77 , 58 , 66 , 79 , 88 , 97 , 86
1) Put the grades in a frequency table with intervals
2) Draw the cumulative frequency polygon
3) Use the graph to estimate the median
Intervals
‫الفئات‬
total
(f)
(cf)
‫) ضع الدرجات في جدول تكراري ذو فئات‬1
‫) ارسم المضلع التكراري التراكمي‬2
‫) استخدم الرسم لتقدر قيمة الوسيط‬3
Grouped Data
The Median Class Interval
The Median Class Interval is the class interval containing the
median.
Example 1.
During 3 hours at Heathrow airport 55 aircraft arrived late. The number of
minutes they were late is shown in the grouped frequency table below.
minutes late
frequency
0-9
27
10 - 19
10
20 - 29
7
30 - 39
5
40 - 49
4
50 - 59
2
(55+1)/2
= 28
The 28th data value is in the 10 - 19 class
Grouped Data
Example 2.
A group of University students took part in a sponsored race. The number of
laps completed is given in the table below. Use the information to:
(a) Calculate an estimate for the mean number of laps.
(b) Determine the modal class.
(c) Determine the class interval containing the median.
Data is
grouped
into 8 class
intervals of
width 4.
number of laps
frequency (x)
1-5
2
6 – 10
9
11 – 15
15
16 – 20
20
21 – 25
17
26 – 30
25
31 – 35
2
36 - 40
1
Grouped Data
Example 2.
A group of University students took part in a sponsored race. The number of
laps completed is given in the table below. Use the information to:
(a) Calculate an estimate for the mean number of laps.
(b) Determine the modal class.
(c) Determine the class interval containing the median.
number of laps
frequency
1-5
2
6 – 10
9
11 – 15
15
16 – 20
20
21 – 25
17
26 – 30
25
31 – 35
2
36 - 40
1
f
 91
midpoint(c.c)
3
8
13
18
23
28
33
38
c.c. X f
6
72
195
360
391
700
66
fx
38
 1828
Mean estimate = 1828/91 = 20.1 laps
Grouped Data
Example 2.
A group of University students took part in a sponsored race. The number of
laps completed is given in the table below. Use the information to:
(a) Calculate an estimate for the mean number of laps.
(b) Determine the modal class.
(c) Determine the class interval containing the median.
number of laps
frequency (x)
1-5
2
6 – 10
9
11 – 15
15
16 – 20
20
21 – 25
17
26 – 30
25
31 – 35
2
36 - 40
1
Modal Class 26 - 30
Grouped Data
Example 2.
A group of University students took part in a sponsored race. The number of
laps completed is given in the table below. Use the information to:
(a) Calculate an estimate for the mean number of laps.
(b) Determine the modal class. 
(c) Determine the class interval containing the median. 
number of laps
frequency (x)
c. F.
1-5
2
2
6 – 10
9
11
11 – 15
15
26
16 – 20
20
46
21 – 25
17
63
26 – 30
25
88
31 – 35
2
90
36 - 40
1
91
The 46th data value is in the 16 – 20 class , median ≈ 18
f
 91
(91+1)/2 =
46