Basic Concepts in Statistics

1. Basic Concepts in Statistics
Mr. Anthony F. Balatar Jr.
Subject Instructor

2. Statistics
• It is a branch of mathematics
mainly concerned with collection,
organization presentation,
analysis and interpretation of
quantitative or numerical data.

3. Two Major Divisions of Statistics
•Descriptive Statistics - are used to
describe the basic features of the data
in a study. They provide simple
summaries about the sample and the
measures. Together with simple
graphics analysis, they form the basis
of virtually every quantitative analysis
of data.

4. Two Major Divisions of Statistics
Descriptive Statistics involves:
-Gathering, classification, organization and
presentation in a form that is
understandable to all.
-Summarize some of the important
features of a set of data.
-Construction of tables and graphs,
computations of measures of locations
and spreads.

5. Two Major Divisions of Statistics
•Inferential Statistics – is used to
make inferences or conclusions about
the population based on sample data.
It is also the process of using data
analysis to deduce properties of an
underlying probability distribution. It
requires a higher order of critical
judgment.

6. Two Major Divisions of Statistics
Inferential Statistics involves:
-Computations for the correlations of the
data.
-Formulate conclusions or generalizations
about a population based on an
observation or a series of observation of a
sample drawn from the population.

7. Population and Samples
•Population – refers to the total
number of people, object or events
that we consider in our study.
•Sample – refers to the collection of
some elements in a population. It
represents the characteristics of a
population.

8. Quantitative VS Qualitative
•Qualitative Variables – it refers to
the attributes or characteristics of a
sample. It is something that is not
measureable but can simply
identified.
•Quantitative Variables – refers to
the numerical values. It is the
numerical information collected about
the samples.

9. Discrete VS Continuous
•Discrete Variables – it results from
either a finite number of possible
values or countable number.
•Continuous Variables – it results
from infinitely many possible values
that can be associated with points on
a continuous scale in such a way that
there are no gaps or interruptions.

10. Level of Measurements
•Nominal level – it is characterized by
data consists of names, labels or
categories only.
•Ordinal Level – it involves data that
may be arranged in some order but
differences between data values either
cannot be determined or are
meaningless.

11. Level of Measurements
•Interval level – these variables
does not only show sameness or
difference of objects or whether
one is less than the other but it
makes statements of equality of
intervals. It does not have a “true-
zero” point, instead it is arbitrarily
assigned.

12. Level of Measurements
•Ratio Level – these are the
variables where the quality of ratio
and proportion is important. This
time, there is a “true-zero” point.
The numbers used represent
distances from a natural origin.

13. Kinds of Data
•Internal data – are those which
are generated from the activities
within the firm.
•External data – are those whose
sources are obtained from outside
the firm.

14. Kinds of External Data
•Primary data – information or
facts which are directly gathered
from the original source.
•Secondary data – the data were
taken from any published or
unpublished materials. These are
most often done through the
method of documentary analysis.

15. Data Collection and
Presentation

16. Methods of Data Collection
•Direct Method – also known as interview
method. A method where there is a person to
person exchange of idea between the one
soliciting information (interviewer) and the one
supplying the data (interviewee). The
researchers may use the structured or
unstructured interview.
- Expensive and time consuming
- Gives more valid result
- Mainly used for a small sample size

17. Methods of Data Collection
•Indirect Method – also known as the paper
and pencil method or the questionnaire
method. Researcher has to prepare questions
relevant to the subject of his/her study.
- Less expensive
- Requires much shorter time
- High possibility of incorrect responses

18. Methods of Data Collection
An indirect method is advised to
have the list of questions conform
with the best feature of writing a
questionnaire and must make sure
that administration is properly
done. It can be mailed to the
respondents or hand carried to the
intended respondents.

19. Methods of Data Collection
•Registration Method – also known as
the documentary analysis where the
researcher make use of the data, fact,
information on file. These documents
are something that is enforced by a
certain law or policy.

20. Methods of Data Collection
•Observation Method – this method is
used if objects of the study cannot talk or
write. Data pertaining to behaviors of an
individual or a group of individuals at the
time of occurrence of a given situation are
best obtain by direct observation. Subjects
maybe taken individually or collectively,
depending on the target of the investigator.

21. Methods of Data Collection
•Experiment Method – this method
examines the cause and effect of
certain phenomena. Data obtained are
done through a series of experiments
which require laboratory result.

22. Features of a Good Questionnaire
• It must be short and clear enough to be understood
by the respondents.
• Avoid stating a leading question.
• Be precise with every statement particularly with the
units to ease the tabulation of data.
• Design a structured questionnaire which can just be
easily checked or blocked by the respondents.
• Limit questions only to the essential information
needed in your study.
• Arrangement and/or sequencing should be properly
done.

23. Sampling Techniques
A. Probability Sampling – it is a sampling procedure
wherein every element of the population is given a
non – zero chance of being selected as a sample.
This is taken to mean that everyone in the
population has the chance to be included in the
sample.
- Simple random sampling
- Systematic sampling
- Stratified sampling
- Cluster sampling
- Multi – stage sampling

24. Probability Sampling
1. Simple Random Sampling – selection is done
fairly, just and without bias. Researcher gives no
criteria or researcher is being objective in the
selection of samples.
2. Systematic Sampling – researcher develops a
certain nth star or simply developing a pattern
which can also be done through random selection.
3. Stratified Random Sampling – can be done by
equal or proportional strata. This is the technique
commonly used particularly if there are several
sources of data.

25. Probability Sampling
4. Cluster Sampling – it is done by choosing samples
in group. When a group is chosen, regardless of
who is in the group, they are all considered as
samples.
5. Multi – Stage Sampling – this technique is referred
to as selection in several stages of sampling.

26. Sampling Techniques
B. Non – Probability Sampling – it is a sampling
technique wherein not every population is given a
chance of being selected as sample. The researcher
states his prejudice for certain samples. These
samples that over – represents or under –
represents some parts of the population is called
biased.
- Purposive Sampling
- Quota Sampling
- Convenience Sampling

27. Non – Probability Sampling
1. Purposive Sampling – it is a non – random
technique of choosing samples where the researcher
defined his criteria or rules. If you meet the criteria
set, then you can be counted as part of the sample.
2. Quota Sampling – the researcher or investigator
limits the number of his samples on the required
number for the subject of his/her study.
3. Convenience Sampling – the researcher chooses
his most preferred location/venue where he conduct
his study. The researcher specifies the place and
time where he can collect his data.

28. Ways to obtain Sample Size
A. By Percentage – for a very large population, 10% of
the population is obtained. For a small population,
20% of the population is desired. This rule seems to
be arbitrary.
B. By Margin of Error – if a researcher wants to have
95% precision in the result of his study, that would
implicate a margin of error of 5%. To solve this, use
Slovin’s Formula: 𝑛 =
𝑁
1 +𝑁𝑒2, where
n = sample size e = margin of error
N = population size

29. Language of Statistics

30. Summation Notation
The symbol 𝒊=𝟏
𝒏
𝑿𝒊 is read as “the summation of x sub
i is from 1 to n”. This is to taken to mean that the
summation goes from 1 to a certain number of n. In
statistics, it is necessary to deal with the sums of
numerical values.
Notice that the summation notation above involved
subscript. A subscript can be a letter or a number
placed at the lower right of a given variable.
Summation uses the Greek alphabet sigma (Σ) which
is taken to mean as the sum of the given items.

31. Laws of Summation
1.Summation of a Constant -
𝑘=1
𝑛
𝑘
2.Summation of a Sum -
𝑖=1
𝑛
(𝑋𝑖 + 𝑌𝑖)
3.Summation of a Variable and
a Constant - [ 𝑘=1
𝑛
(𝑋𝑖 + 𝑘]

32. Data Presentation

33. Types of Data Presentation
1. Textual Presentation – Data collected is presented
in paragraph form if it is purely qualitative or when
there are very few numbers involved. This method is
commonly adopted by researchers undergoing
qualitative research.
2. Tabular Presentation – the more effective way of
presenting data which appears in the form of rows
and columns. It can be easily for comparison and
emphasis. It has four major components: table
heading, body, stubs and box heads.

34. Types of Data Presentation
3. Graphical Presentation – it is presented in visual
form. It may appear in many forms: line, bar, circle
and picture graphs.
a. Line Graph - it is an effective device to show the
changes in values with respect to time and is
plotted in the rectangular coordinate system. It
can sketch through straight line, dotted line or
broken line to show relationship between two or
more set of quantities.

35. Types of Data Presentation
b. Bar Graph – it is commonly used to illustrate
data and make easy comparisons between sets of
data.
- simple bar chart
- component bar chart
- composite bar chart
c. Circle Graph – it is drawn to represent the whole
quantity. The circle is then divided into a few sectors
to show the relative magnitude between the
components of the given quantity.

36. Types of Data Presentation
c. Circle Graph – the area of each sector is
proportional to the magnitude of the component it
represents.
The angle of each sector is:
𝑴𝒂𝒈𝒏𝒊𝒕𝒖𝒅𝒆 𝒐𝒇 𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕
𝑴𝒂𝒈𝒏𝒊𝒕𝒖𝒅𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒘𝒉𝒐𝒍𝒆
𝒙 𝟑𝟔𝟎°
d. Pictograph – it is used to dramatize the
differences among the few quantities. In this
method, pictorial symbols are used to represent
data. Simple pictorial symbols can give an
immediate visual impact on readers. However,
pictographs cannot give accurate information.

37. Frequency Distribution

38. Frequency Distribution
•It is the tabular arrangement of
data by classes or categories
together with their corresponding
frequencies.

39. Steps in Constructing Frequency Distribution
1. Find the range of values. Get the difference of the
highest value (HV) and the lowest value (LV).
2. Determine the desired class interval. The ideal
number of class intervals (CI) is somewhere between
5 and 15 preferably odd class intervals. But a more
scientific way is by applying the formula:
CI = 3.33 + log n
3. Compute for the class size (i). Divide the
computed range (R) by the desired computed class
interval (CI). i = R/CI

40. Steps in Constructing Frequency Distribution
4. Construct a frequency table by making class
intervals. Starting with the lowest value in the
lower limit of the first class interval, then add the
computed class size to obtain the lower limit of the
next class interval. Continue adding the class size
on the lower limits until you reach the desired class
interval.
5. Determine the number of data (frequency) for
every class interval by tallying the raw data.
6. Write the obtained frequency (f) from each class
interval by counting the tallied form.

41. Steps in Constructing Frequency Distribution
7. Determine the class mark (x) of each class
interval. Add the lower limit (LL) and the upper
(UL) then divide the sum by 2 to get its midpoint.
8. Determine the class boundaries (CB) or class
limits. Subtract 0.5 from every lower limits and add
0.5 from every upper limits.
9. Determine the cumulative frequency less than
(<cf) and the cumulative frequency greater than
(>cf).
10.Obtain the relative frequencies (RF) to
determine the percentage distribution of
frequencies.

42. Graphical Representation
of Frequency Distribution

43. Steps in Constructing Frequency Distribution
1. Frequency Polygon – it is a line graph of class
frequencies plotted against the class mark.
2. Histogram – it is a series of columns, consisting of
a set of rectangles having bases on a horizontal axis
which center on the class mark.
3. Ogive – it is a graphical representation of
cumulative frequencies. The graph of less than ogive
is a rising frequency polygon while the graph of
greater than ogive is a falling frequency polygon.
The intersection of two ogives is called the median.

44. Measures of Central
Tendency

45. MEASURES OF CENTRAL
TENDENCY
Three Measures of Central Tendency: (Ungrouped Data)
1. Mean – it indicates a point around which the values in the distribution
balance.
Formula: 𝑿 =
𝑋 𝑖
𝑁
where 𝑿 = mean, Xi = scores,
𝑿𝒊 = sum of the scores N = total frequency

46. MEASURES OF CENTRAL
TENDENCY
Three Measures of Central Tendency: (Ungrouped Data)
1. Mean – it indicates a point around which the values in the distribution
balance. (Weighted Mean)
Formula: 𝑋 =
𝑓𝑋
𝑁
where 𝑿 = mean, f = frequency,
X = score 𝒇𝑿 = sum of the product of frequency and score
N = total frequency

47. MEASURES OF CENTRAL
TENDENCY
Three Measures of Central Tendency: (Ungrouped Data)
2. Median ( 𝑋)– it is the value in the distribution which divides an arranged
(ascending or descending) the distribution into two equal parts.
Formula: 𝑋 = [(N + 1) / 2]th position
3. Mode ( 𝑋) – it is the number that occurs most often in a data set.

48. MEASURES OF CENTRAL
TENDENCY
Three Measures of Central Tendency: (Grouped Data)
1. Mean – (Weighted Mean)
Formula: 𝑿 =
𝑓𝑋 𝑚
𝑁
where 𝑿 = mean, f = frequency,
Xm = class mark (average of lower interval and upper interval)
𝒇𝑿 = sum of the product of frequencies and class marks
N = total frequency

49. MEASURES OF CENTRAL
TENDENCY
Three Measures of Central Tendency: (Grouped Data)
1. Mean – (Coded Deviation Method)
Formula: 𝑋 = 𝑋 𝑜 +
𝑓𝑋 𝑐
𝑁
𝑖 where 𝑿 = mean, f = frequency,
Xc = coded value
𝑋 𝑚−𝑋 𝑜
𝑖
N = total frequency
𝒇𝑿 = sum of the product of frequencies and class marks

50. MEASURES OF CENTRAL
TENDENCY
Three Measures of Central Tendency: (Grouped Data)
2. Median 𝑋 = 𝑋 𝐿𝐵 +
𝑁
2
− 𝑐𝑓 𝑏
𝑓 𝑚
𝑖
𝑿 = median 𝑿 𝑳𝑩 = lower boundary or true lower limit of the median class
N = total frequency cfb = cumulative frequency before the median class
fm = frequency of the median class i = size of the class interval

51. MEASURES OF CENTRAL
TENDENCY
Three Measures of Central Tendency: (Grouped Data)
3. Mode 𝑋 = 𝑋 𝐿𝐵 +
∆1
∆1+∆2
𝑖 𝑿 = mode i = size of the class interval
𝑿 𝑳𝑩 = lower boundary or true lower limit of the modal class
∆ 𝟏= difference between the frequency of the modal class and the frequency of the class
interval preceding it
∆ 𝟐= difference between the frequency of the modal class and the frequency of the class
interval succeeding it

52. Measures of Position

53. MEASURES OF POSITION
Quantiles – is referred to as the division of items in the
distribution into equal parts.
a. Quartiles – it is referred to as the division of items into four equal
parts.
b. Deciles – it is referred to as the division of items into ten equal parts.
c. Percentiles – it is referred to as the division of items into one hundred
equal parts.

54. Measures of Variation
and Dispersion

55. MEASURES OF VARIATION
Measures of Variation and Dispersion: (Grouped Data)
1. Range – it is defined as the difference between the highest
score (h.s.) and the lowest score (l.s.) – ungrouped
Range – it is defined as the difference between the upper
boundary (u.b.) and the lower boundary (l.b.) – grouped
Range = h.s – l.s. = u.b. – l.b.

56. MEASURES OF VARIATION
Measures of Variation and Dispersion: (Grouped Data)
2. Interquartile Range (I.R.) – it is the difference between the
75th percentile or Q3 and the 25th percentile or Q1.
Thus, IR = Q3 – Q1
3. Quartile Deviation (Q. D.) – it is one half the value of the
interquartile range. Thus, Q. D. = IR/2

57. MEASURES OF VARIATION
Measures of Variation and Dispersion: (Grouped Data)
4. Mean Absolute Deviation (M. A. D.) – it is equal to the
average, for a set of numbers , of the differences between
each number and set’s mean value. Thus,
M.D. =
|𝑋 − 𝑋|
𝑁
or M.D. =
𝑓|𝑋 𝑚 − 𝑋|
𝑁

58. MEASURES OF VARIATION
Measures of Variation and Dispersion:(Ungrouped Data)
5. Variance (S2) and Standard Deviation(S) –
S2 =
(𝑋 − 𝑋)2
𝑁 − 1
or S.D. =
(𝑋 − 𝑋)2
𝑁 − 1

59. MEASURES OF VARIATION
Measures of Variation and Dispersion:(Grouped Data)
5. Variance (S2) and Standard Deviation(S) –
S2 =
𝑁 𝑋2 −( 𝑋)
2
𝑁(𝑁 − 1)
or S.D. =
𝑁 𝑋2 −( 𝑋)
2
𝑁(𝑁 − 1)
S2 =
𝑁 𝑓𝑋2
𝑚 −( 𝑓𝑋 𝑚)
2
𝑁(𝑁 − 1)
or S.D. =
𝑁 𝑓𝑋2
𝑚 −( 𝑓𝑋 𝑚)
2
𝑁(𝑁 − 1)