
A branch of Applied Mathematics
specializing in procedures for collecting,
organizing, presenting, analyzing, and
interpreting data from observations.
 Statistics involves much more than simply
drawing graphs and computing averages.
 In education, it is frequently used to
describe test results.
 In science, the data resulting from
experiments must be collected and
analyzed.
 Manufacturers can provide better
products at reasonable costs through
the use of statistical quality control
techniques.
 In government, many kinds of statistical
data are collected all the time.
 A knowledge of statistics can help you
become more critical in your analysis of
information; hence, you will not be
misled by manufactured polls, graphs,
and averages.
Statistical Question → one that can be
answered by collecting data that vary.
Example of Statistical Questions:
1.
How many hours do college students
spend time in studying? (summarizing
question)
2.
Do college students spend more time in
social media than studying? (comparing
question)
3.
Do students who spend more time in
studying do better in exam? (relationship
question)
Example of Non-Statistical Questions:
1.
How old are you?
2.
What is your favorite subjects?
It is the analysis of data that help to
describe, show, and summarize data
under study
Organize, analyze and present data in a
meaningful way
It is used to describe a situation
It explain already known data and limited
to a sample and population having small
size
Types: Measure of central tendency &
Measure of variability
Results are shown with the help of charts,
graphs, tables, etc.
Commonly Used Summarizing Values

Percentage

Measures of central tendency and
location

Measures of variability

Skewness and Kurtosis
Examples:
1) Class average of examination
2) Range of students’ score
3) Average salary
It is the analysis of random sample of
data taken from a population to describe
and make inference about the population
Compares, test, and predict data
It is used to explain the chance of
occurrence of an event
It attempts to reach the conclusion about
the population
Types: Estimation of parameters &
Testing of hypothesis
Results are shown with the help of
probability scores
Commonly Used Statistical Tools or
Techniques

Estimation of Parameters

Testing of hypothesis

(z-test, t-test, ANOVA, Chi-squares,
regression, Time series analysis)
Examples:
1) Significant relationships between job
satisfaction and performance of CCDC
employees
2) The use of module is significantly
effective than the traditional method of
teaching
Two types of data:
1. QUANTITATIVE DATA - numerical
values
2. QUALITATIVE DATA - categorical
responses such as colors, information, or
questions that are answerable by YES or
NO, labels, genders, attitude, etc.
QUANTITATIVE
Discrete
continuous
- numerical characteristics or
attributes associated with the population that
can assume different values.
-
is collection of facts or information.
- finite number of values;
values are obtained by counting (number
of students, number of passers, etc.)
2. Continuous - infinite number of
values between two specific numbers;
values are obtained by measuring (weight,
height, temperature)
Levels of Measurement
Nominal - classifies and categorizes
data
Ex: type of blood, gender, religion,
citizenship
Ordinal - rank or order to show
relationship
Ex: President (officers), eldest (family
order)
Ratio - value of zero or starts at an
absolute zero point
Ex: mass, length, time, angle, energy,
rating, electrical change, test results
Interval - variables are measured
based on a set of intervals on a certain
scale
Ex: temperature (freezing point, boiling
point)

1. Discrete
- this method is
used when the objective is to determine
the cause and effect relationship of a
certain phenomenon under controlled
condition.
SAMPLING TECHNIQUES
POPULATION includes all of the elements
from a set of data; objects, events,
organizations, countries, species, organisms,
etc.
SAMPLE is a subset taken from a population,
either by random sampling or by non-random
sampling
A. RANDOM SAMPLING

Selection of n elements derived from
the N population, which is the subject of
an investigation or experiment, where
each point of the sample has an equal
chance of being selected using the
appropriate sampling technique.
Types of Random Sampling Techniques
1.
Lottery Sampling - each member of the
population has an equal chance of being
selected.
Systematic sampling - members of the
population are listed and samples are
selected at intervals called sample
intervals. In this technique, every nth item
in the list will be selected from a randomly
selected starting point.
 Ex: If you want to draw a 200 sample
from a population of 6,000, select
every 3rd person in the list.

3.
Stratified random sampling - members
of the population are grouped on the
basis of their homogeneity.
 This technique is used when there are
number of distinct subgroups in the
population within which full
representation is required.
 The sample is constructed by
classifying the population into
sub-populations or strata on the basis
of certain characteristics of the
population, such as age, gender or
socioeconomic status.
2.
COLLECTION OF DATA
Statistical of Data

it focuses in determining
the charges in the attitude, characteristics
and behavior of people or other subjects.
This technique includes watching and
recording actions and behaviors. The
person who gathers the data is called an
investigator while the person being
observed is called the subject.

- oral or verbal
communication where the interviewer
asks questions in any mode (face to face,
telephone, or virtual) to an interviewee.

- gathered through a set
of question that is mailed or handed to
respondents who are expected to read
and understand them.

- if you have a big number pf
samples, it is the most practical way to
use, in a national level, surveys are
usually covered by the government and
other forms of surveying organization
such as Philippine Statistic Authority
(PSA).
Example: Select a sample of 400 students
from the population which are grouped
according to the cities they came from. The
table shows the number of students per city.
City
Population (N)
A
B
C
D
Solution: To determine the number of
students to be taken as sample from each city,
we divide the number of students per city by
total population (N=28,000) multiply the result
by the total sample size (n=400).
City
A
B
C
D
Population (N)
12,000
10,000
4,000
2,000
Total= 28,000
Sample (n)
x (400) = 171
x (400) = 143
x (400) = 57
x (400) = 29
Total= 400
12,000
28,000
10,000
28,000
4,000
28,000
2,000
28,000
Cluster Sampling - applied on a
geographical basis. Generally, first
sampling is performed at higher levels
before going down to lower levels.
4.

12,000
10,000
4,000
2,000
Some forms of graphs for ungrouped
frequency distributions are pie hart, bar
graph, and line graph
PIE CHART (Pie Graph)
Used to show how all the parts of
something are related to the whole.
It is represented by a circle divided into
slices or sectors of various sizes that
show each part’s relationship to the whole
and to other parts of the circle.


BAR GRAPH
Uses rectangles (or bars) of uniform width
to represent data, particularly the nominal
or categorical type of data. The height of
the rectangle denotes the frequency of
the variable.
Two types of bar graph:

- sometimes called a
column chart. Used to show the
changes in the numerical value of a
variable over a period of time.


Example: Samples are taken randomly from
the provinces first, follows by cities,
municipalities, or barangays, and then from
households.
Multi-stage Sampling - uses a
combination of different sampling
techniques.
5.

Example: Selecting respondents for a national
election survey, use the lottery method first for
regions and cities, then use stratified sampling
to determine the number of respondents from
selected areas and clusters.
- used to represent
changes in data over a period of time.
Data like changes in temperature,
income, population, and the like can
be represented by a line graph. Data
are represented by points and are
joined by line segments. A line graph
may be curved, broken, or straight.
DATA refers to information that is collected
and recorded. It can be in the form of numbers,
words, measurement, and much more.
Grouped Data is the type of data which is
classified into groups after collection.
Ungrouped Data which is also known as raw
data that has not been placed in any group or
category after collection.

When data are presented as graph, they are
easily interpreted and compared.

Data in an ungrouped frequency
distribution can be presented
graphically to give a better picture of
the distribution.

Some forms of graphs for grouped
frequency
distribution
are
the
Histogram and Ogive.
HISTOGRAM
A bar graph that shows the frequency of
data that occur within a certain interval. In
a histogram, the bars are always vertical,
the width of each bar is based on the size
of the interval it represents, and there are
no gaps because their bases cover a
continuous range of possible values.
Ie for a given ungrouped data to be
transformed as a group data.
Example: the following are the test scores of
students. Construct a suitable frequency table.
Use 6 as the desired number of class interval.

OGIVE
Also called the
cumulative
frequency
graph or
cumulative
frequency curve
is a graph
plotted from a cumulative frequency table.
 Frequency Distribution Table
Frequency - the number of occurrence of a
data
Frequency Table - a table that lists items and
shows the number of times the items occur.

Steps in constructing a frequency table
(for ungrouped data)
Step 1: Make 3 columns. Arrange the data in
order in the first column.
Step 2: Make a tally.
Step 3: Count the tallies then write the
frequencies.
Step 4: Total all the frequencies

Steps in constructing a frequency
distribution table
1. Determine the range. Range is the
difference between the highest value H and
the lowest value L in the set of data. R = H - L
2. Determine the desired number of the class
interval or categories. The ideal number of
class interval in somewhere between 5 and
15.
3. Determine the class width or approximate
size of the class interval by dividing the range
by the desired number of class intervals.
Class Width =
𝑅𝑎𝑛𝑔𝑒
𝐶𝑙𝑎𝑠𝑠 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑠
W=
𝑹
𝑪𝒍
4. Write the class intervals starting with the
lowest lower value as determined in the data.
Then add the class width to the starting point
to get the next interval. Do this until the
highest value is contained in the last interval.
5. Tally the corresponding number of scores in
each interval. Then summarize the results or
sum up the tallies under the frequency column.
14
15
30
26
30
10
10
30
34
20
30
10
22
18
14
21
19
11
Solution:
1. Determine the range.
19
15
40
22
26
16
10
15
20
36
17
29
18
28
40
36
37
R=H-L
= 40 - 10
= 30
2. Class interval = 6
3. Determine the Class Width
W=
𝑹
𝑪𝒍
=
𝟑𝟎
𝟔
=5
4. Write the class interval starting with the
lowest lower value as determined in the data.
- starting with 10 and with W = 5, the
class intervals are: 10 - 15, 16 - 21, 22 - 27, 28
- 33, 34 - 39, 40 -45.
5. Tally the corresponding number of scores in
each interval. Then summarize the results or
sum up the tallies under the frequency column.
Scores
10 - 15
16 - 21
22 - 27
28 - 33
34 - 39
40 - 45
Tally
IIII IIII
IIII IIII
IIII
IIII I
IIII
II
TOTAL
Frequency
10
9
4
6
4
2
35
DATA → refers to information that is collected
and recorded. It can be in the form of
numbers, words, measurement and much
more.
UNGROUPED DATA → which is also known
as raw data that has not been placed in any
group or category after collection.
GROUPED DATA → the type of data which is
classified into groups after collection.
Measures of Central Tendency of
Ungrouped Data
MEAN or the Arithmetic Mean

Most commonly used measure of central
position.

It is used to describe a set of data where
the measures cluster or concentrate at a
point.
∑𝒙
̅ =
𝒙
𝑵
 Where ∑ 𝑥 = the summation of 𝑥 (sum
of the measures
 𝑁 = number of values of 𝑥
Example: The grade in Probability of 10
students are 87, 84, 85, 85, 86, 90, 79, 82, 78,
76. What is the average grade of the 10
students?
Sulotion:
∑𝒙
̅ =
𝒙
𝑵
𝑥̅ = 87 + 84 + 85 + 85 + 8610+ 90 + 79 + 82 + 78 _76
𝑥̅ = 832
10
𝑥̅ = 83.2
 Hence, the average grade of the 10
students is 83.2.
MEDIAN

Middle value or term in a set of data
arranged according to size/magnitude
(either increasing or decreasing)

For data with two median, add the two
middle values and divide it by two.
Example 1: The library logbook shows that 58,
60, 54, 35, and 97 books, respectively, were
borrowed from Monday to Friday last week.
Find the median.
Solution:

Arrange the data in increasing order. 35,
54, 58, 60, 97

Since the middle value is the median,
then the median is 58.
Example 2: Andrea’s scores in 10 quizzes,
during the first quarter are 8, 7, 6, 10, 9, 5, 9, 6,
10, and 7. Find the median.
Solution:

Arrange the data in increasing order. 5, 6,
6, 7, 7, 8, 9, 9, 10, 10

Since the number of measures is even,
then the median is the average of the two
middle score.
Md = 7 +2 8 = 7.5

Hence, the median of the set of scores is
7.5.
MODE

The measure of value which occurs most
frequently in a set of data
 The value with the greatest frequency.
 To find the mode for a set of data:
1. Select the measures that appears most
often in the set:
2. If two or more measures appear the same
number of times, then each of these
values is a mode” and
3. If every measure appears the same
number of times, then the set of data
has no mode.
MEASURES OF VARIABILITY
(UNGROUPED DATA)
Measure of dispersion or variability
 refer to the spread of the values about the
mean.
 These are important quantities used by
statistician in evaluation. Smaller
dispersion of score arising from the
comparison often indicates more
consistency and more reliability.
 The most commonly used measures of
dispersion are the range, the average
deviation, the standard deviation, and
the variance.
1.

-- the difference between the
largest value and the smallest value.
R=H-L
Where R = range, H = highest value, L =
lowest value
Example: Test score of 10, 8, 9, ,7, 5, and 3.
R=H-L
R = 10 - 3
R=7
2.
-- the
dispersion of a set of data about the
average
̅|
A.D. = ∑|𝒙−𝒙
𝑵
Where A.D is the average deviation;
𝑥 is the individual score;
𝑥̅ is the mean; and
𝑁 is the number scores
|𝒙 − 𝒙
̅| is the absolute value of the
deviation from the mean.
Procedure in computing the average deviation:
1) Find the mean for all the cases.
2) Find the absolute difference between
each score and the mean.
3) Find the sum of the difference and divide
by 𝑁.
𝑁 is the number scores.
Example: Find the average deviation of 12, 17,
13, 18, 15, 14, 17, 11.
1)
Find the mean 𝑥̅ .
∑𝑥
𝑥̅ =
= 12 + 17 + 13 + 18 +9 15 + 14 + 17 + 11
𝑁
135
𝑥̅ = 9 = 15
2)
Find the absolute difference between
each score and the mean.
|𝑥 − 𝑥̅ | = |12 − 15| = 3
= |17 − 15| = 2
= |13 − 15| = 2
= |18 − 15| = 3
= |18 − 15| = 3
= |15 − 15| = 0
= |14 − 15| = 1
= |17 − 15| = 2
= |11 − 15| = 4
= |12 − 15| = 3
3)
Find the sum of the absolute difference
∑|𝑥 − 𝑥̅ |.
|𝑥 − 𝑥̅ |
= |12 − 15| = 3
= |17 − 15| = 2
= |13 − 15| = 2
= |18 − 15| = 3
= |18 − 15| = 3
= |15 − 15| = 0
= |14 − 15| = 1
= |17 − 15| = 2
= |11 − 15| = 4
= |12 − 15| = 3
∑|𝑥 − 𝑥̅ |
=
20
x
12
17
13
18
18
15
14
17
11
𝑥̅
15
15
15
15
15
15
15
15
15
|𝑥 − 𝑥̅ |
3
2
2
3
3
0
1
2
4
Example: compute the standard deviation of
the set of test scores: 39, 10, 24, 16, 19, 26,
29, 30, 5.
1)
Find the mean.
2)
Find the deviation from the mean (𝑥 − 𝑥̅ ).
3)
Square the deviations (𝑥 − 𝑥̅ )2 .
4)
Add all the squared deviations ∑(𝑥 − 𝑥̅ )2 .
5)
Tabulate the results obtained:
6)
Compute the standard deviation (SD)
using the formula
̅)
SD = √∑(𝒙−𝒙
𝑵
𝟐
(𝑥 − 𝑥̅ )2
𝑥
5
10
16
19
24
26
29
30
39
∑(𝑥 − 𝑥̅ )2
4.
-- the variance ꝺ2 of a
data is equal to 1⁄𝑁 .
ꝺ𝟐 =
∑(𝒙−𝒙
̅ )𝟐
𝑵
Where ꝺ2 is the variance;
𝑁 is the total number of observations;
x is the raw score; and
𝑥̅ is the mean of the data.

Variance is not only useful, it can be
computed with ease, and it can also be
broken into two or more component sums
of squares that yield useful information.
∑|𝑥 − 𝑥̅ | = 20
4)
Solve for the average deviation by dividing the result
in step 3 by 𝑁 .
A.D. =
3.
∑|𝑥−𝑥
̅|
𝑁
=
20
9
∑(𝑓𝑥)
∑𝑓
= 2.22
Median : 𝑙𝑏𝑚𝑐 + [
-differentiates sets of scores with equal
averages. But the advantage of standard
deviation over mean deviation is that it
has several applications in inferential
statistics.
̅)
SD = √∑(𝒙−𝒙
𝑵
Mean : 𝑥̅ =
∑𝑓
2 − <𝑐𝑓
𝑓𝑚𝑐
]i
i = interval
𝑙𝑏 = lower boundary → LL - 0.5
𝑐𝑓 = cumulative frequency
𝑓𝑚𝑐 = frequency of the median class
mc = median class → ∑2𝑓 = 40
= 20th
2
𝟐
Where SD is the standard deviation;
𝑥 is the individual score;
𝑥̅ is the mean; and
Mode : 𝑙𝑏𝑚𝑜 + [
2
𝐷1 +𝐷2
]i
mo = modal class
= highest frequency
𝐷1 = difference
𝐷2 = difference
𝑥
41-45
36-40
31-35
26-30
21-25
16-20
𝑖=5
𝑓
1
8
8
14
7
2
𝑥̅
43
38
33
28
23
18
∑ = 40
𝑓𝑥
43
304
264
392
161
36
∑ = 1,200
1,200
Mean =
40
𝑥̅ = 30
40
−9
Median = 25.5 + [ 214 ]5
25.5 + [11
]5 = 29.43
14
Mode = 25.5 + [6 +6 7]5
6
25.5 + [13
]5 = 27.81
𝐷1 = 14 - 8 =
𝐷2 = 14 - 7 =
6
7
𝑙𝑏
40.5
35.5
30.5
25.5
20.5
15.5
< 𝑐𝑓
40
39
31
23
9
2