Introduction to Statistics
By
Ismaila Zango Mohammed
Department of Sociology
Bayero University, Kano
1. Introduction
Statistics is of value because we are constantly exposed to statistics in our every day life. For
instance voting polls, results of consumer survey are reported in news papers. In addition to that
you may be interested in reading research reports and interpreting them. Quite often we tend to
associate our world with numbers such as birth rate, unemployment rate, and divorce rates.
Statistics is applicable to a wide variety of academic disciplines from the natural and social
sciences to the humanities, government and business. In this chapter attempt is made to define
statistics, differentiate descriptive from inferential statistics. Examples of the two classifications
are provided with illustrations. The chapter also deals with measures of central tendency and
dispersion.
Statistics is a mathematical science dealing with the collection, analysis, interpretation or
explanation, and presentation of data. In other words, Statistics is a branch of mathematics that
deals with the collection, organization, and analysis of numerical data. According to FrankNachmias and Leon-Guerrero (2000) statistics refers to a set of procedures used by social
scientists to summarize, organize and communicate information represented in numbers. This
type of information is called data. Similarly, Agretis and Franklin (2007) define statistics as the
art and science of designing studies and analyzing data that those studies produced. In other
words the main aim of statistics is to translate data into knowledge and understanding of the
world around us. In addition it also encompasses prediction and forecasting based on data.
From the last definition three are things have emerged, which include design, description, and
inference. Design means planning how to gather data to answer the questions of interest. Thus,
statistics is a reliable means of describing accurately the values of economic, political, social,
psychological, biological, and physical data and serves as a tool to correlate and analyze such
data. Uses of statistics are no longer confined to gathering and tabulating data, but are chiefly a
process of interpreting the information.
1.1 Descriptive Statistics
Descriptive statistics are branch of statistics that help researchers to summarize, organize and
describe data collected from population or sample. Example of descriptive statistics include
frequency, percentage, measures of central tendency and measures of dispersion, which allow the
researcher to give a vivid description of events, population and distribution of properties in the
population. According Trochim (2001) descriptive statistics are used to describe the basic
features of the data in a study. They provide simple summaries about the sample and the
1
measures. Together with simple graphic analysis, they form the basis of virtually every
quantitative analysis of data. Descriptive statistics is simply used to describe what is going on in
our data. Descriptive Statistics are used to present quantitative descriptions in a manageable
form. In a research there are many variables to measure, or may involve lager number of people
on a particular measure. For instance in 2007 post election survey in northwest zone in Nigeria a
sample 4820 was drawn. We may be interesting in knowing how many of these people have
voted, how many of them are males/females and the age distribution of the sample. This can be
done by knowing the distribution of the sample by age, sex and voting. In other words the data
can be organized according distribution, similarity of certain features for example age (central
tendency) of differences (variation or dispersion). There are certain measures, which will be
discussed below
1.2 Inferential Statistics
Inferential statistics on the other hand are concerned with making prediction or inference about a
population from observations and analysis of samples. In others words, from the observation of a
sample, inferences can be made about the entire population with regards to the variables the
researchers is interest in. With inferential statistics according to Trochim (2001) you are trying to
reach conclusions that extend beyond the immediate data alone. For instance, we use inferential
statistics to try to infer from the sample data what the population might think. Or, we use
inferential statistics to make judgments of the probability that an observed difference between
groups is a dependable one or one that might have happened by chance in this study. Thus, we
use inferential statistics to make inferences from our data to more general conditions. In other
wards inference is a process of making generalization of drawing conclusion about attributes of a
population from evidence from the sample
According to Knoke, Bohrnstedt and Mee (2002) there are number of statistical significant test
that allow for making inference that conclusion drawn from a sample are true for the population
from which the sample was drawn. The similarity of the sample based observation and the
population to a large extent depends on random selection of sample. Random selection of sample
ensures that every element in the population has an equal chance of being part of the sample.
Examples of inferential statistics include t test, Z test, correlation, regression etc.
2. Types and Sources of Statistical Data
Data according to Marshal (1998) are records of observation, which can take many forms such as
scores in IQ tests, interview records, field diaries. In other words, data are information, often in
the form of facts or figures obtained from experiments or surveys, used as a basis for making
calculations or drawing conclusions. Statistical data can be classified into primary and
secondary. Data in general could also be categorized into quantitative and qualitative.
Quantitative data is any information represented by numbers that can be subjected to statistical
2
analysis, while qualitative data are facts collected which may not necessarily be converted into
numbers. According to Sambo (2008) qualitative data consist of transcripts of individual
interviews, focus group discussion, field notes, schemes of work, photographs etc. Both
qualitative and quantitative could classified as either primary or secondary
Primary data are collected by the researcher directly from the subjects using questionnaire or
structured interview. Secondary data exist, usually generated for reasons other than reasons of
the current researcher. For example a researcher interested in the quality of housing in Nigeria
can use data from 2006 census. Secondary data can be qualitative or quantitative. However, for
this chapter we are concerned with quantitative data, the decision to use secondary data depends
on whether the existed data is of good quality. Sometimes the existing data may even be of
superior quality. Secondary data are obtainable in Nigeria from government agencies such as
National Population Commission. National Bureau of Statistics, National Planning Commission,
Central Bank of Nigeria etc. There are also other databases such Uniform Crime Reports, reports
from school enrollment, data on marriage and divorce, vital statistics (record of births and
deaths). There are also international databases produced by United Nations agencies such as
United Nation Development Programme (UNDP), United Nations International Children
Education Fund (UNICEF), World Bank etc.
3. Discussion
3.1 Distribution.
This is the way and manner particular characteristics researchers are interested in shared across
the sample. For instance how many people in the sample are males or females? This information
can be presented in terms of the number of occurrence in the sample (frequency). This can be
presented in frequency table as follows. Distributions of discrete variable classify persons,
objects or events according to quality of their attributes for instance level of education, while
continuous variable classify them according their quantities. Frequency distribution is a table of
outcome or response category of variable and the number of time each outcome is observed.
Distribution according to Nachmias and Nachmias (1987) is the first step in data analysis and
makes sense in relation to other frequencies. Frequencies expressed in comparable number are
called proportion or percentages and are usually expressed as:
Proportion = fi/N
Percentage =fi/N x 100
For instance using data in Table 1 the proportion of male to female is fi/N = 3032/4820 = 0.629
and the percentage is fi/N x 100 = 3032/4820 x 100 =0.629 x 100 = 62.9
Table 1 below is an example of frequency distribution table. The information can also be
presented in charts; the data in table1 is presented in pie chart.
3
Table 1 Sex distribution of Respondents
Sex
Male
Female
Total
Frequency
3032
1788
4820
Proportion
0.629
0.371
1.00
Percent
62.9
37.1
100
Source: Centre for Democratic Research and Training (2007)
Chart 1 Sex Distribution of Respondents
Percent
37.1
Male
62.9
Female
The frequency counts could be transformed into relative frequency of proportion by dividing the
number of cases in each outcome by the total number of cases. To determine the proportion of
male in table 1 we simple divide number of male by the total number of cases 3032/4820 = 0.629
the proportion is presented in column 3 in table 1. This allows us to compare the number males
to female in the sample. The proportion could be transformed into percentages by multiplying
the proportion by 100. For example the percentage of females is 0.371 x 100 = 37.1. The
percentage is standardize for sample and are usually presented in the nearest tenth because it will
make sense to talk of one tenth of a person. Values of 0.1- 0.4 are rounded down while numbers
from 0.5- 0.9 are rounded upward (Knoke, Bohrnstedt and Mee 2002)
In statistical language notations and shorthand are often used represent things. For instance N
refer to the total number of cases, f denoting frequency associated th outcome (category) of
variable. The subscript can take from 1 to the numbers of categories K. In this case our K is 2
that is the number of categories. The distribution of the sample can be presented in tabula or
graphic form such as chart.
i
i
i
4
3.2 Measure of Central Tendency
In addition to the distribution of the sample, data often exhibit a cluster or central point, this
number is called a measure of central tend we may be interested in a having single figure that
summarizes the information about our sample that is the central tendency, or average value of set
of scores. It is important to however, note that the central tendency is only meaningful at the
interval level of measurement. The measures of central tendency include the mode, median and
mean.
i. The mode (MO) is the most frequent number in the distribution and thus it is easy to
determine. For instance thirty students gave their age as follows 15, 15, 17, 17, 18, 18,
19, 20, 20, 21, 22, 22, 23, 23, 23, 24, 24, 27, 28, 29, 29, 30, 30, 31, 32, 35, 36, 36, 37, 38.
In this age distribution our mode is 23, which appeared more than any figure in the
distribution. It is important to however note that some distribution could have more than
one mode. Such distributions are known as bimodal distribution. In the case of grouped
data the category that has the highest frequency is the model category. If the data above is
grouped as indicated below. The model category is age group 20-24, which has the
highest frequency
For grouped data the mode is the mid-point of the category that has the largest number of
cases for example the mode in Table 2 is located in age category of 20-24 years, which
has the largest frequency. To find the mid point we simply add lower and upper limit of
the response category, which in this case is 20+24/2= 22. The mid point is 22, thus the
mode in this case is easy to determine.
Table 2 Age Distribution
Age category
15-19
20-24
25-29
30-34
35-39
Total
Freq
7
10
4
4
5
30
Percent
23
33
13
13
17
99
ii. The Median (Mdn) is applicable to variable with categories that can be arranged in a
sequence from lowest to highest. In other words to find the median the cases have to be
arranged from the lowest to the highest or from highest to lowest. The median is the
outcome that divides the distribution exactly into half. That is half of the cases will be
below the median and the other half will be above the median. For the above data on age
the median is 23.5. The existence of the median depends on whether the distribution is
even or odd. In the case of odd distribution, there is one middle number below, which
half of the cases are found and over which the other half are found. For example consider
these two age distributions.
Distribution A: 15, 15, 17, 17, 18, 18, 19, 20, 20
5
Distribution B: 15, 15, 17, 17, 18, 18, 19, 20, 20, 21, 22, 23
Distribution A has an odd number of cases (nine). Thus, its median score is the fifth
observation or 18. Distribution B has even number of cases (twelve) so it median falls half
between the sixth and the seventh cases; therefore its value is the average of the two cases
(18+19)/2 = 18.5. In other words, the median is the middle case for odd number
distribution of the average scores for the two middle cases for even number distribution.
For grouped data however, the median is value of that category at which the cumulative
percentage reaches 50.0% (Knoke, Bohrnstedt and Mee 2002)
The median for the grouped data is computed with assumption that cases in the category
containing the median are evenly distributed throughout the interval. The formula for
computing the median for grouped data is
L + 1/2N-F W
f
mdn
Where:
L = lower true limit of the category containing the median/lower class limit of the
median class
N= Total number of cases
F=Cumulative frequency up to but not including the frequency of the median interval
f = The frequency of the median class
W = Width of interval containing the median or size of the median class
mdn
To obtain the true limit we divide the difference between adjacent limits by 2. To obtain
the true limit In case of table 4, 30-29 = 1/2 =0.5. Subtract this from lower limits and add
it to the upper limits to get the true limit.
Table 4 Computation of the Median for Grouped Data
Stated Limit
True Limit f
F
20-29
19.5-29.5
7
7
30-39
29.5-39.5
3
10
40-49
39.5-49.5
9
19
50-59
49.5-59.5
4
23
60-69
59.5-69.5
2
25
To obtain our median, we need to locate the median interval. The median interval will be
the one with the middle number. If the data is ungrouped, we will simply rank the 25
cases and pick the middle case. However, we cannot do this for grouped data. In this case
we should locate the interval that contains the middle case. Since N is 25, the middle case
is N/2 = 12.5. Thus we are looking for the interval that contains twelfth and thirteenth
cases. If we look at our cumulative frequencies, we will notice that the interval containing
6
the twelfth and the thirteenth cases is 39.5-49.5. This is our median interval. The l
be the lower of this interval. Using our formula we will have
mdn
will
39.5+ 25/2-10 x 10
9
= 39.5+ 12.5-10 x10
9
= 39.5+ 2.5 x 10
9
= 39.5+2.8 = 42.3
Our median for this grouped data is 42.3 marks.
iii. The Mean the Arithmetic Mean (AM) and often referred as the average, is the
commonest measure of central tendency. The mean is calculated for continuous
distributions and interval variables. The mean is arrived at by adding all the numbers of
observations and dividing by total number of cases. The A M is obtained for data not in
frequency table, which is simply the average of a given number of data The mean is
obtained using the following formula
 = ∑X
N
Where
∑ = Summation
X = observation
N = Number of case
The formula requires us to add all the observations from the first case to the last and then
divide by the number of cases. In order to illustrate the computation of the mean, apply
the formula to the data of the thirty students presented below:
A.M = (15+ 15+ 17+ 17+ 18+ 18+ 19+ 20+ 20+ 21+ 22+ 22+ 23+ 23+ 23+ 24+ 24+ 27+
28+ 29+ 29+ 30+ 30+ 31+ 32+ 35+ 36+ 36+ 37+ 38)/30 = 759/30 = 25.3
The arithmetic mean age for this distribution is 25.3.
For data in a frequency distribution the mean is calculated using the following formula
 = ∑fx
∑f
7
Where
∑ = summation
f = frequency
x= observation
With grouped data the formula for calculating the mean is different and slightly more complex
as presented below:
 = ∑fi mi
∑fi
Where
∑ = Summation
f = frequency of interval
m = mid point of the interval
i
i
Table 5 Computation of the Mean for Grouped Data
Stated Limit
True Limit Mi
fi
20-29
19.5-29.5
24.5
7
30-39
29.5-39.5
34.5
3
40-49
39.5-49.5
44.5
9
50-59
49.5-59.5
54.5
4
60-69
59.5-69.5
64.5
2
Total
25
fi mi
171.5
103.5
400.5
218.0
129.0
1022.5
Note 1: The mid point is obtained by adding the lower and upper limits (either of stated or true
limit) of each interval and dividing the sum by 2. For instance for the first interval in table 5 the
mid point will be :
20+29/2 = 49/2 = 24.5
Note 2: The fi mi is obtained by multiplying the mid point by the frequency of the interval for
example to obtain the fi mi for the second interval see illustration below:
34.5 x 3 = 103.5
To obtain the  we simply divide the sum of fi mi by the sum of fi which is illustrated below
 = ∑ fi mi = 1022.5/25 = 40.9
∑ fi
8
3. 3 Measure of Dispersion/Variability
Researchers are frequently concerned with the variability of the distribution, that is, whether the
measurements are clustered tightly around the mean or spread over the range. Measures of
dispersion include range, mean deviation, variance and standard deviation.
i. The range is the simplest measure of dispersion or variability. A distribution range is
defined as the difference between the largest and the smallest score. The range of age in
our earlier example is 15 to 38 years or 38-15 = 23. The range is 23 that is the difference
between least age and the highest age in the distribution.
ii. Percentile is score below which a specific percentage of the distribution falls. Percentile
are used to evaluate relative performance on standardize test such University
Matriculation Examination (UME). The nth percentile is a score below which n percent
of the distribution falls. For instance, the 75 percentile is a score that divide the
distribution so that 75 percent of the cases are below and 25 percent are above it. The
median is usually the 50 percentile, which is a score that divide the distribution is such a
way that 50 percent (or half) of the cases fall below it. Percentiles are meaningful for data
that is at ordinal or higher level of measurement. Percentiles are easy to identify if the
data are arranged in a frequency distribution One measure of this variability is the
difference between two percentiles, usually the 25 low values and the 75 percentiles
(high values).
th
th
th
th
iii. The Mean Absolute Deviation (MAD): This is simply the sum of the difference
between the each of the score and the mean divide by the number of cases. It is important
to note that absolute value of the number is the value of x without the negative sign and is
written as ( / x /) for instance /-2/ = 2 In summing the differences the signs are ignored
the formula is MAD = ∑/X-/
N
In order to calculate the MAD we need to obtain the mean. The mean for the data below
is  = ∑X = 95+61+71+50+90+77+81/7 = 525/7 = 75
N
Table 6 Computation of the Mean Absolute Deviation
State
X
X-
/X-/
JG
95
20
20
KN
61
-14
14
KT
71
-4
4
KD
50
-25
25
KB
90
15
15
SK
77
2
2
ZM
81
6
6
Total
525
0
86
9
To obtain the MAD (∑/X-/ /N) using the formula is simply to subtract the mean from
the X value for instance in case the first entry in Table 6, 95-75 = 20. The sum of /X-/
divide N.
MAD = ∑/X-/ = 86/7 = 12.3
N
iv. The Variance is the square of the standard deviation. It is a measure of variation for
interval-ratio variables and is the average of squired deviations from the mean. For
example we may be interested in determining the variation of poverty across the seven
states in the northwest zone. To calculate the variance we need to find the mean, calculate
the difference of each of the scores to the mean and the squired sum of the difference.
The formula for raw date is presented below:
Ѕ = ∑(x-)
N
2
Whereas
∑= summation sign
X= Scores
= Mean
N= number of observation
Table 7 Showing the Poverty Rate in Northwest Zone
State
X
X-
(X-)
JG
95
20
400
KN
61
-14
196
KT
71
-4
16
KD
50
-25
625
KB
90
15
225
SK
77
2
4
ZM
81
6
36
Total
525
0
1502
Source: Ogwumike (2006)
2
 =527/7 =75
The formula for calculating variance for grouped data is presented below:
Ѕ = ∑(fi x )
2
N
10
Table 5 Computation of the Variance for Grouped Data
Age category
fi
mi
fi mi x= mi-
15-19
7
17
119
-8
20-24
10
22
220
-3
25-29
4
27
108
2
30-34
4
32
64
7
35-39
5
37
185
12
Total
30
696
The mean for the data is 25.3 approximated to 25
x
64
9
4
49
144
2
/fi xi/
56
30
8
28
60
182
fi x
448
90
16
196
720
642
2
For grouped data some adjustments have to be made to obtain the variance. The
average deviation from the mean becomes ∑/fi x//N. Where fi is interval frequency
and Xi= mi - or each interval mid point minus the mean. Thus our variance for the
data in Table 5 is:
∑(fi x ) = 642/30 = 21.4
2
N
v.
The Standard Deviation (SD) is a measure of variability that is more scientific than mean
deviation for further investigation and analysis of statistical data. The SD is defined as
the square root of the variance. The formula for raw data is presented below
___________
SD= √∑(x-) /N
2
In the previous example the SD;
SD = √1502/7 = √214.57 = 14.65
3.4 Skewness
The distribution of the data can be described by the shape of the general distribution, which can
be visually presented in a curve. A distribution according to Frank-Nachmias and Leon-Guerrero
(2000) can either be symmetrical or skewed depending on whether there are a few extreme
values at one end of the distribution. A distribution is said to be symmetrical if the frequencies at
the right and left tails are identical. In other words, if the distribution is divided into two
halves, each will be a mirror image of the other. Example of symmetrical distribution is
presented in Table 6. Symmetry or lack of it in a distribution is determined using coefficient of
skewness. The coefficient of skewness is defined as:
11
Coefficient of Skewness = mean- mode or
SD
Or
3(maen – median)
SD
SD = Standard Deviation
Table 6 Hypothetical Income Distribution (Symmetrical)
Income (X)
Frequency
FX
1
N 1,000
N 1,000 ☺
2
N 4,000
N 2,000 ☺ ☺
4
N 12,000
N 3,000 ☺ ☺☺☺
2
N 8,000
N 4,000 ☺ ☺
1
N 5,000
N 5,000 ☺
Total
10
N 30,000
The mean income for this distribution is  = ∑fx/∑f = 30,000/10 = N 3,000. The median is also
N 3,000. The distribution clusters around the mean
However in a skewed distribution there are few extreme values on one side of the distribution. A
distribution with extreme low values is said to be negatively skewed, while a distribution with
few extreme high values is referred to as positively skew. In a skewed distribution the mean is
pulled in the direction of the extreme values (either extreme low or extreme high scores).
Example of skewed distribution is presented in Table7.
Table 7 Hypothetical Income Distribution (Positively Skewed)
Income (X)
Frequency
FX
1
N 1,000
N 1,000 ☺
2
N 4,000
N 2,000 ☺ ☺
4
N 12,000
N 3,000 ☺ ☺☺☺
2
N 8,000
N 4,000 ☺ ☺
N 50,000 ☺
Total
1
N 50,000
N 75,000
10
The mean income for this distribution is  = ∑fx/∑f = 75,000/10 = N 7,500. The median is
N3,000 and the mode is also N3,000. In this distribution the mean is affected by the extreme
high value, thus the mean is N6,000 while the median is still N 3,000. The value of the mean is
twice the value of the median and the mode in this case. In order to determine the degree of
skewness in distribution in table we need to calculate the SD, which is 5.18. To calculate the
coefficient of skewness we will the following:
Mean =
Mode =
N 6,000
N 3,000
12
SD
=
5.18
Coefficient of Skewness = mean- mode = 6000-3000 = 3000 = 579.2
SD
5.18
5.18
The coefficient of skewness can either be a positive or negative value, which indicates the nature
of the skewness. Higher the value of the coefficient indicates the magnitude of the skewness. In
our example the value is 579.2, which shows that the data is positively skewed.
4. Summary Conclusion
In the chapter some commonly used methods of organizing, summarizing data and finding the
centre of ungrouped and grouped data have been treated. The methods include mode, median and
mean. Similarly, some measures of dispersion have also been treated such as range, mean
average deviation, variance and standard deviation. The distribution helps researchers to
summaries, organize and display data. Difference and trends within group can be identified using
simple frequency distribution table. The mean in the absence of extreme scores (outliers) is the
most recommended. On the other hand, standard deviation is the most scientific measure of
dispersion compared to the mean average deviation. Depending on the data the distribution can
be symmetrical or skewed. In general, statistics is a useful tool for summarizing, presenting and
analysis of data.
13
Exercises
1. Thirty students were asked about their age and the following data were obtained
17, 18, 22, 20, 20, 22, 23, 30, 23, 30, 31, 32, 29, 35, 37, 36, 19, 21, 23, 24, 36, 38, 15, 18,
17, 24, 27, 28, 29
a. Construct frequency table using class interval of 5, taking into account the lower
limit of 15.
b. Compute the percentage and cumulative percent for each category
2. From the data Table 1 below compute the percentage of those currently married and
those ever married
Table 1 Marital Status of Respondents
Marital Status Frequency
Married
710
Single
226
Divorced
35
Separated
06
Widowed
37
Source: Department of Sociology, Bayero University, Kano (2009)
3. From the data in table 2 compute the percentage of those with secondary level of
education.
4. Calculate cumulative frequency and percentage.
Table 2 Respondents Level of Education
Level of Education
No Schooling
Qur’anic Eduction
Adult Literacy
Primary
Secondary
Tertiary
Source: Mohammed (2001)
5.
Frequency
21
407
71
146
245
269
Using data in table 3 calculate the median age at first married using formula for grouped
data
14
Table 3 Age at First Marriage
Age at Marriage
15-20 Years
21-25 Years
26-30 Year
31-36 Years
Total
Source: Mohammed (2001)
Frequency
351
429
246
74
1100
6. Using data in table 3 calculate the mean age using formula for grouped data
Poverty rate in Southwest of Nigeria
Ekiti 42%
Ondo 42%
Lagos 64%
Oyo 24%
Osun 32%
Ogun 32%
Source (Ogwumike, 2006)
7. From the data above what is the range of poverty in southwest of Nigeria
8. From the data above calculate Mean Deviation, Variance and Standard Deviation
Table 4 Respondents Desired Total Number of Children
Desired
Total
Number of Children
1-5
6-10
11-15
16-20
21-25
26-30
Source: Mohammed (2001)
Frequency
57
53
8
9
6
1
9. Using the data in Table 4 calculate the variance using grouped data formula.
10. Using the data in Table 4 calculate the standard deviation using grouped data formula.
15
References
Agretis, A and Franklin, C (2007) Statistics: The Art and Science of Learning From Data. New
Jersey: Pearson Prentice Hall
Babbie, Earl (2004) The Practice of Social Research (Belmont CA: Wadsworth/Thompson)
Centre for Democratic Research and Training (2007) Educating the Electorate: A Survey Report
on Voter Education Project and Public Opinion on Nigeria’s 2007 General
Election in Northwest Geo-political Zone of Nigeria, Submitted to UNDP,
Abuja, Nigeria
Department of Sociology, Bayero University, Kano (2009) Strategies For Enhancing Community
Security & Conflict Resolution Mechanisms In Katsina State A Report
Submitted To Safety And Security Component Manager DfID, 3 Floor, AP
Plaza, Adetokunbo Ademola Crescent, Wuse II, Abuja-Nigeria
rd
Frankfort-Nachmias, C and Leon-Guerrero, A. (2000) Social Statistics for Diverse Society
(Thousand Oak CA: Pine Forge Press)
Knoke, D. Bohrnstedt, G. W. and Mee, A. P. (2002) Statistics for Social Data Analysis. Belmont:
Wadworth and Thompson
Marhal, G (1998) Oxford Dictionary of Sociology, Oxford: Oxford University Press
Mc Tavish, D and Loether, H, (1999) Social Research. Longman
Mohammed, I.Z. (2001) Male Attitudes Toward and Use of Family Techniques in Kano State A
PhD Thesis Submitted to the Department of Sociology, Bayero University,
Kano.
Ogwumike, F (2006) Nigeria Poverty Assessment: Quantitative Aspect. Draft report Submitted
to the United Nations Development Programme, Abuja
Nachmias, D and Nachmias, C (1987) Research Methods in the Social Sciences (New York: St
Martins Press)
Trochim, William M K (2001) Research Methods Knowledge Base. Cincinnati, OH : Atomic
Dog Publishers., 2001
Sambo, A. A. (2008) Research Methods in Education Lagos: Stirling-Horden Publishers Ltd.
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