Principles of Statistics
Descriptive Statistics
Frequency Table (Distribution)
Scenario 1:
The President of the university called the
registrar and asked for the number of students
in each faculty classified by admission year and
sex. How the registrar will prepare the required
information?
Scenario 2:
The dean of the faculty of science asked the
chair of the Mathematics department to submit a
list of the names of the top 5% of the graduates
of the year 2010. How this can be done?
Frequency Table of Discrete Data
To construct a frequency table of a given
discrete data set apply the following steps:
1. Find the different categories of the data
2. Partition the data set by categories
3. Find the frequency of each category
4. Give a title to the table
Example 1.5.1: Consider the following data:
A, B, AB, A, B, O, A,B, A,B, AB, AB, O,A, B, AB, A, O, B,
B, A, A, A, O
Solution:
Note that the different observations in this data are A, B, AB,
and O. Hence the required table is
Distribution of Blood Type
Class
Frequency
A
9
B
7
AB
4
O
4
Sum
24
Frequency Table of Continuous Data
Example 1.5.4: Consider the following data
34, 56, 45, 34, 23, 12, 23, 34, 55, 66, 77, 88, 99,
90, 45, 56, 65, 78, 87, 98, 89, 23, 12, 21, 32,
35, 48
Class
Frequency
12-33
7
34-55
8
56-77
5
78-99
7
Sum
27
Computations of the Previous Measures
Class
Mid(mi)
f
m.f
(m^2). F
8-9
8.5
2
17
144.5
9-10
9.5
4
38
361
10-11
10.5
10
105
1102.5
11-12
11.5
5
57.5
661.25
12-13
12.5
3
37.5
468.75
13-14
13.5
1
13.5
182.25
268.5
2920.25
Midpoint=(lower+upper)/2,
f=frequency
mf
X   n  268.5 10.74
25
2
m2 f

2
S  n  X 116.81115.35 1.46
Coding:
U=(x-10.5)/1
-2
-1
0
1
2
3
f
2
4
10
5
3
1
25
u. f
-4
-4
0
5
6
3
6
(u^2). f
8
4
0
5
12
9
38
mean of U=6/25=0.24,
variance of U=(38/25)-(0.24)^2=1.46
Percentiles:
Upper Class Limit
Cumulative Frequency
9.5
2
10.5
6
11.5
16
12.5
21
13.5
24
14.5
25
(1) Median: Rank=(25)(0.5)=12.5
Smallest change in freq.=6.5, Largest change=10
Median=10.5+(6.5/10)(1)=11.15
(1=length of the class)
(2) First Quartile - Rank=25(0.25)=6.25
Smallest change in freq.=0.25, Largest change=10
Q1=10.5+(0.25/10)*(1)=10.525.
(3) 70th Percentile - - - > Rank=25(0.7)=17.5
Smallest change=1.5, Largest change=5
70th Perc.=11.5+(1.5/5)(1)=11.8
Two Problems:
1. The mean of 5 numbers is 4 and Std. is 2. Two numbers were 2, 2
are changed into 3, 4. Find the mean and Std. of the new set.
Old mean =4 =(sum of 3 numbers+4)/5 - - -> sum of 3 numbers=16
New mean=(16+7)/5=23/5=4.6
Old var. =4=(sum of squares+8-(5*16))/4 - - - > sum of squares=88
New var. =(88+9+16-(5)(4.6^2))/4=(113-105.8)/4=7.2/4=1.8
New Std. =1.34
2. A list contains 9 numbers having mean 6 and Std. 5. If the number
16 is added to the list, find the mean and std. of the new list.
Sum of the data = 54.
New mean=(54+16)/10=7
Old var. =25=(sum of squares-(9*36))/8=(sum of squares-324)/8
- - -> sum of squares=524.
New var. =(524+256-(10)(49))/9=(780-490)/9=290/9=32.22
New Std. =5.68