Statistics and statistical methods pdf.

STATISTICS
INTRODUCTORY CONCEPTS

STATISTICS AND STATISTICAL
METHODS
Statistics is a body of knowledge that deals with the
collection, presentation, analysis and interpretation of
numerical and categorical data.
Statistical methods refers to the procedures used in the
collection, presentation, analysis and interpretation of data.

WHY STUDY STATISTICS?
The field of statistics is the science of learning from data.
Statistical knowledge helps you use the proper methods to
collect the data, employ the correct analyses, and effectively
present the results. Statistics is a crucial process behind how we
make discoveries in science, make decisions based on data, and
make predictions. Statistics allows you to understand a subject
much more deeply.

TWO AREAS OF STATISTICS
Descriptive Statistics comprises statistical methods concerned
with collecting and describing a set of data so as to yield
meaningful information.
Inferential Statistics comprises statistical methods concerned
with the analysis of a subset of data leading to predictions or
inferences about the entire set of data.

POPULATION AND SAMPLE
Population consists of the totality of observation which we are concerned
about.
Sample is a subset of a given population .
Parameter refers to any numerical value describing a characteristic of a
population.
Statistic refers to any numerical value describing a characteristic of a
sample.

EXAMPLE
Situation: There are twelve major suppliers of Calamansi fruit across
Region 8. They were surveyed, and data on the amount of their monthly
Calamansi fruit supply for the past seven years were gathered from each
supplier. The following were part of the results of the survey:
a. The average monthly supply of the twelve major suppliers for the past
seven years is 24.90 thousand metric tons.
b. The top 1 supplier is Supplier C whose average monthly supply for the
past seven years is 6.80 thousand metric tons.
c. There will be an increase of Calamansi fruit supply by the major
suppliers in the coming years.

EXAMPLE
What comprises the population?
Monthly amount of Calamansi fruit supply for the past seven years
of each of the twelve major suppliers
Give a sample mentioned in the given situation.
Monthly amount of Calamansi fruit supply for the past seven years
of Supplier C
Give an example of parameter.
Average monthly supply of the twelve suppliers for the past seven
years (24.90 thousand metric tons)

EXAMPLE
Give an example of statistic.
Average monthly supply of Supplier C for the past twelve years
(6.80 thousand metric tons)

VARIABLE AND CONSTANT
Variable is a characteristic of objects or individuals that can take on
different values for different members of the group under study.
Example: IQ, height and weight of all 10-year old female children
in a particular barangay
Constant is a characteristic that assumes the same value for all
members of the group.
Example: Age, sex and home address of all 10-year old female
children in a particular barangay

TYPES OF VARIABLES
Quantitative Variable is a variable that takes only numerical values.
Example: I.Q., height, weight, income and age
Qualitative Variable is a variable that takes only non-numerical
values, and numbers are used only as categories
Example: Gender, sex, religion, year level, educational attainment
and occupation

TYPES OF VARIABLE ACCORDING TO
LEVEL OF MEASUREMENT
Nominal – numbers are used merely as labels of the categories of
the variable
Ex. sex, religion, and occupation
Ordinal – have the same characteristic with a nominal variable and
in addition, the numbers can be meaningfully ranked
Ex. economic status, year level, and salary grade
Interval – have the same characteristic with an ordinal variable and
in addition, the categories in the interval scale are defined in terms

TYPES OF VARIABLE ACCORDING TO
LEVEL OF MEASUREMENT
of a “standard unit of measurement” so that equality of differences
between successive categories of the scale is defined
Ex. temperature in Degree Celsius and IQ score
Ratio – have the same characteristic with an interval variable and in
addition, it has a true zero point
Ratio and interval variable are what you call scale variables while
ordinal and nominal variable are categorical variables.
Ex. age, height, and number of siblings

FREQUENCY DISTRIBUTION
It is a tabular arrangement of data indicating the different classes
or categories and the corresponding frequencies.
There are three basic types of frequency distributions. The three
types are categorical, ungrouped and grouped frequency
distributions.

CATEGORICAL FREQUENCY
DISTRIBUTION
The categorical frequency distribution is used for data that can be
placed in specific categories, such as nominal- or ordinal-level data.
For example, data such as political affiliation, religious affiliation, or
major field of study.

EXAMPLE
Twenty-five army inductees were given a blood test to determine
their blood type. The data set is as follows:
RAW DATA OF BLOOD TYPE
A B B AB O
O O B AB B
B B O A O
A O O O AB
AB A O B A

EXAMPLE
Blood Type Tally Total %
A 5 25 20%
B 7 25 28%
O 9 25 36%
AB 4 25 16%

UNGROUPED FREQUENCY
DISTRIBUTION
An ungrouped frequency distribution is used for numerical
data and when the range (the difference between the highest
and the smallest values) is small.

EXAMPLE
Consider the following table, which lists the number of laptop computers owned
by families in each of 40 homes in a subdivision.

EXAMPLE
No. of Laptops Tally Total %
0 5 40 12.5
1 12 40 30
2 14 40 35
3 3 40 7.5
4 2 40 5
5 3 40 7.5
6 0 40 0
7 1 40 2.5

EXAMPLE
No. of Laptops Frequency Percentage
0 5 12.5
1 12 30
2 14 35
3 3 7.5
4 2 5
5 3 7.5
6 0 0
7 1 2.5
N=30 100

GROUPED FREQUENCY
DISTRIBUTION
When the range of the data is large, the data must be grouped into classes that
are more than one unit in width.
To construct a frequency distribution, follow these rules:
1. There should be between 5 and 20 classes.
2. The class width should be an odd number. This ensures that the midpoint of
each class has the same place value as the data.
3. The classes must be mutually exclusive. Mutually exclusive classes have
nonoverlapping class limits so that data cannot be placed into two classes.
4. The classes must be continuous. There should be no gaps in a frequency
distribution.

GROUPED FREQUENCY
DISTRIBUTION
When the range of the data is large, the data must be grouped into classes that
are more than one unit in width.
To construct a frequency distribution, follow these rules:
5. The classes must be exhaustive. There should be enough classes to
accommodate all the data.
6. The classes must be equal in width. This avoids a distorted view of the data.

GROUPED FREQUENCY
DISTRIBUTION
CONSTRUCTING GROUPED FREQUENCY DISTRIBUTION
1. Find the range.
𝑟𝑎𝑛𝑔𝑒 = ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 − 𝑙𝑜𝑤𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒
2. Decide on the number of class intervals or classes, we denote it by 𝑘.
➢𝑘 = 𝑛
➢5-20 classes
3. Determine the class size or class width of the interval, we denote it by c.
(rounded to the nearest odd whole number)
4. Determine the lower limit LL and the upper limit and the upper limit UL of the lowest
class interval. The lowest class interval should contain the lowest value in the data set.

GROUPED FREQUENCY
DISTRIBUTION
The value of the UL is determined using the equation.
𝑈𝐿 = 𝐿𝐿 + (𝑐 − 1)
5. Determine the upper class intervals by consecutively adding the class size 𝑐 to
the values of LL and UL of the lowest class interval until we get the class interval
with the highest value in the data set.
6. Tally the data, find the frequencies.

GROUPED FREQUENCY
DISTRIBUTION
▪The class boundaries are used to separate the classes so tat there are no
gaps in the frequency distribution.
Rule of Thumb: Class limits should have the same decimal place value as the
data, but the class boundaries have one additional place value and end in a 5.
▪The class midpoint is found by adding the upper and lower boundaries (or
limits) and dividing by 2.
▪The cumulative frequencies are used to determine the number of cases failing
below (for <cf) or above (for >cf) a particular value in a distribution.

GROUPED FREQUENCY
DISTRIBUTION
▪The relative frequency (rf) of a class interval is the proportion of observations
falling within the class and maybe presented in percent. Thus,
𝑟𝑓 =
𝑓
𝑛
× 100

EXAMPLE
Distribution of scores of forty students in a Mathematics class.
RAW SCORES
76 92 87 78
87 88 85 92
67 85 93 91
85 79 92 82
99 95 79 85
81 96 75 88
82 86 83 87
79 92 80 74
86 93 98 71
81 86 80 94

EXAMPLE
SOLUTION:
Step 1: 𝑅𝑎𝑛𝑔𝑒 = 𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 − 𝐿𝑜𝑤𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 = 99 − 67 = 32.
Step 2: 𝑘 = 𝑛 = 40 ≈ 7
Step 3: 𝑐 =
32
7
≈ 5
Step 4: 𝑈𝐿 = 65 + 5 − 1 = 65 + 4 = 69
Step 5: 𝑈𝐿 = 70 + 5 − 1 = 74
𝑈𝐿 = 75 + 5 − 1 = 79
𝑈𝐿 = 80 + 5 − 1 = 84

EXAMPLE
SOLUTION:
𝑈𝐿 = 85 + 5 − 1 = 89
𝑈𝐿 = 90 + 5 − 1 = 94
𝑈𝐿 = 95 + 5 − 1 = 99

EXAMPLE
SOLUTION:
Step 6:
Class Interval 𝒇 𝒙 𝒓𝒇 < 𝒄𝒇 > 𝒄𝒇
95-99 4 97 10 40 4
90-94 8 92 20 36 12
85-89 12 87 30 28 24
80-84 7 82 17.5 16 31
75-79 6 77 15 9 37
70-74 2 72 5 3 39
65-69 1 67 2.5 1 40

HISTOGRAM, FREQUENCY
POLYGONS AND OGIVES
The histogram is a graph that displays the data by using contiguous vertical
bars of various heights to represent the frequencies of the classes. (class
boundaries along 𝑥-axis)
The frequency polygon is a graph that displays the data by using lines that
connect points plotted for the frequencies, at the midpoints of the classes. (class
midpoints along 𝑥-axis)
The ogive (cumulative frequency graph) is a graph that shows the cumulative
frequencies for the classes. (with connected points and class boundaries along 𝑥-
axis)

HISTOGRAM, FREQUENCY
POLYGONS AND OGIVES

RELATIVE FREQUENCY GRAPHS
Class Interval 𝒇 𝒓𝒇 < 𝒄𝒇 > 𝒄𝒇 𝒄𝒓𝒇
95-99 4 10% 40 4 10%
90-94 8 20% 36 12 30%
85-89 12 30% 28 24 60%
80-84 7 17.5% 16 31 77.5%
75-79 6 15% 9 37 92.5%
70-74 2 5% 3 39 97.5%
65-69 1 2.5% 1 40 100%

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