Statistics and
Probability
Quarter 3 – Module 7:
Percentiles and t-distribution
Statistics and Probability
Alternative Delivery Mode
Quarter 3 – Module 7: Percentile and T-Distribution
First Edition, 2020
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Statistics and
Probability
Quarter 3 – Module 7:
Percentiles and T-Distribution
Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our dear learners,
can continue your studies and learn while at home. Activities, questions, directions,
exercises, and discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-bystep as you discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each
SLM. This will tell you if you need to proceed on completing this module or if you
need to ask your facilitator or your teacher’s assistance for better understanding of
the lesson. At the end of each module, you need to answer the post-test to self-check
your learning. Answer keys are provided for each activity and test. We trust that you
will be honest in using these.
In addition to the material in the main text, Notes to the Teacher are also
provided to our facilitators and parents for strategies and reminders on how they can
best help you on your home-based learning.
Please use this module with care. Do not put unnecessary marks on any part
of this SLM. Use a separate sheet of paper in answering the exercises and tests. And
read the instructions carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answering the
tasks in this module, do not hesitate to consult your teacher or facilitator.
Thank you.
What I Need to Know
After going through this module, you are expected to:
1. Illustrate the t-distribution (M11-12SP-IIIg2)
2. Identify the percentile using the t-table (M11-12SP-IIIg5)
1
What I Know
A. Read the statements carefully and choose the letter of the best
answer. Write your answer on a separate sheet of paper.
1. What is the other term for t-distribution?
A. Z-distribution
B. Probability distribution
C. Percentile distribution
D. Student’s t-distribution
2. This refers to the maximum number of logically independent values which
varies in the data sample.
A. Level of significance
B. Percentiles
C. Degree of freedom
D. Probability
3. This refers to the measure of position whose data is divided into 100 parts.
A. Percentiles
B. Quartiles
C. Deciles
D. Median
4. What is the level of significance if the confidence interval is 90% ?
A. 0.05
B. 0.10
C. 0.005
D. 0.25
5. What will be the degree of freedom of the data with a sample size of 20?
A. 17
B. 18
C. 19
D. 20
For number 6 to 8, use the given data below to answer the following questions.
“There are 9 students who will be taking up the test, the researcher sets the level of
significance to 0.05. The following are the scores of nine students in the test:
23, 25, 25, 26, 27, 28, 30, 40, 45
2
6. What will be the t-value of the given problem?
A. 1.729
B. 1.456
C. 1.612
D. 1.551
7. Using the t-table, give the value of 25th percentile.
A. 25
B. 26
C. 27
D. 28
8. What will be the t-value of 95th percentile?
A. 1.729
B. 1.456
C. 1.612
D. 1.551
9. What will be the degree of freedom with a t-value of 1 and a sample size of 2?
A. 1
B. 2
C. 3
D. 4
10. Find the t-value of 2.5th percentile on the t-table with a degree of freedom of
6.
A. 2.570
B. 4.030
C. -2.570
D. -4.030
II. For numbers 11-15, complete the table below. Write your answer on a separate
sheet of paper.
Percentile
40th percentile
75th percentile
90th percentile
95th percentile
97.5th percentile
n
23
20
25
15
10
3
t
(11)
(12)
(13)
(14)
(15)
Lesson
1
Percentile and the
T-Distribution
Percentiles have been discussed on your Grade 10 and their uses. These were
also discussed that percentiles are identifier of a student’s performance in
comparison to the performance of the whole class.
In this lesson, you will explore and understand the student’s t-distribution
and identify the percentiles using the t-tables.
Before we discuss percentiles and t-distribution, it is necessary to study first
the concept of percentiles. Let us have a recall about percentiles. Answer all the
activities provided for you. Good luck and have a good time answering the module.
What’s In
A. Using the given data below, find the following percentiles.
40, 45, 50, 50, 56, 60, 65, 80, 90
th
1. 10 percentile
2. 25th percentile
3. 50th percentile
4. 75th percentile
5. 95th percentile
B. Answer the following questions.
1. If Peter is higher than the 75 % of the class, what is his percentile rank?
2. If the 50th percentile of the scores of the class is 80, what will be the median?
3. If the scores of the 3 students are 45, 50, and 60, what will be the 25th
percentile?
4. If the scores are 20, 30, 50, 60, 70, 75, and 80, what will be 75th percentile?
5. The percentile rank of Rose is 50, what percent of the class is her score
belongs?
4
Notes to the Teacher
This part aims to assess if the students have prior knowledge
about the topic. Also, it prepares the students to absorb the
lesson.
What’s New
Find 5 words on the puzzle below. For each word you find, make a brief definition
or expectation about the word.
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1. Using the words that you have found, what are the things that come into
your mind?
5
2. What are the possible things or applications of these words in statistics?
3. Recalling your topic in normal distribution, how will it be related to the
normal curve?
The answer in this question requires an understanding of t-distribution.
What is It
The t-distribution (also called Student’s t-distribution) is a family of
distributions that look almost identical to the normal distribution curve, only a bit
shorter and stouter. The t-distribution is used instead of the normal distribution
when you have small samples. The larger the sample size, the more the t distribution
looks like the normal distribution. In fact, for sample sizes larger than 20 (e.g. more
degrees of freedom), the distribution is almost exactly like the normal distribution.
The t-distribution is similar to a normal distribution. It has a precise
mathematical definition. Instead of diving into complex math, let’s look at the useful
properties of the t-distribution and why it is important in analysis of any set of
statistical quantitative data.

Like the normal distribution, the t-distribution has a smooth shape.

Like the normal distribution, the t-distribution is symmetric. If you think about
folding it in half at the mean, each side will be the same.

Like a standard normal distribution (or z-distribution), the t-distribution has a
mean of zero.

The normal distribution assumes that the population standard deviation is known.
The t-distribution does not make this assumption.

The t-distribution is defined by the degrees of freedom. These are related to the
sample size.

The t-distribution is most useful for small sample sizes, when the population
standard deviation is not known, or both.

As the sample size increases, the t-distribution becomes more similar to a normal
distribution.
Using the t-table, shown on the next page we can be able to identify the t value of a
certain data.
6
Let us be familiar first with the following words:
1. Degree of freedom - This refers to the maximum number of logically
independent values which vary in the data sample.
2. Percentile – This is a measure of position with data divided into 100 parts.
Below is the t-table which you can use to find the t-value
df/v
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
0.40
0.325
0.289
0.277
0.271
0.267
0.265
0.263
0.262
0.261
0.260
0.260
0.259
0.259
0.258
0.258
0.258
0.257
0.257
0.257
0.257
0.257
0.256
0.256
0.256
0.256
0.256
0.256
0.256
0.256
0.256
0.255
0.254
0.254
0.25
1.000
0.816
0.765
0.741
0.727
0.718
0.711
0.706
0.703
0.700
0.697
0.695
0.694
0.692
0.691
0.690
0.689
0.688
0.688
0.687
0.686
0.686
0.685
0.685
0.684
0.684
0.684
0.683
0.683
0.683
0.681
0.679
0.677
0.10
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.303
1.296
1.289
0.05
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.684
1.671
1.658
right tailed area ( α )
0.025
0.01
0.005
12.706 31.821 63.657
4.303
6.965
9.925
3.182
4.541
5.841
2.776
3.747
4.604
2.571
3.365
4.032
2.447
3.143
3.707
2.365
2.998
3.499
2.306
2.896
3.355
2.262
2.821
3.250
2.228
2.764
3.169
2.201
2.718
3.106
2.179
2.681
3.055
2.160
2.650
3.012
2.145
2.624
2.977
2.131
2.602
2.947
2.120
2.583
2.921
2.110
2.567
2.898
2.101
2.552
2.878
2.093
2.539
2.861
2.086
2.528
2.845
2.080
2.518
2.831
2.074
2.508
2.819
2.069
2.500
2.807
2.064
2.492
2.797
2.060
2.485
2.787
2.056
2.479
2.779
2.052
2.473
2.771
2.048
2.467
2.763
2.045
2.462
2.756
2.042
2.457
2.750
2.021
2.423
2.704
2.000
2.390
2.660
1.980
2.358
2.617
7
0.0025
127.321
14.089
7.453
5.598
4.773
4.317
4.029
3.833
3.690
3.581
3.497
3.428
3.372
3.326
3.286
3.252
3.222
3.197
3.174
3.153
3.135
3.119
3.104
3.091
3.078
3.067
3.057
3.047
3.038
3.030
2.971
2.915
2.860
0.0010
318.309
22.327
10.215
7.173
5.893
5.208
4.785
4.501
4.297
4.144
4.025
3.930
3.852
3.787
3.733
3.686
3.646
3.610
3.579
3.552
3.527
3.505
3.485
3.467
3.450
3.435
3.421
3.408
3.396
3.385
3.307
3.232
3.160
0.0005
636.619
31.599
12.924
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
4.015
3.965
3.922
3.883
3.850
3.819
3.792
3.768
3.745
3.725
3.707
3.690
3.674
3.659
3.646
3.551
3.460
3.373
Illustrative example:
1. Identify the t-value whose number of samples n = 7 and has an area (𝛼)
equal to 0.05.
2. Find the t-value whose degree of freedom is 20 and has 𝛼 = 0.01.
3. Identify the t-value of the following percentiles given the number of samples.
a. 95th percentile , N= 30
b. 90th percentile, N=20
Solution:
1. To identify the t-value, identify first the degree of freedom using the formula
df= n-1 where n is the sample size.
df = n-1
df = 7-1
df = 6
Locate the t-value on the t-table using the degree of freedom and the area (𝛼)
df/v
1
2
3
4
5
6
7
0.40
0.325
0.289
0.277
0.271
0.267
0.265
0.263
0.25
1.000
0.816
0.765
0.741
0.727
0.718
0.711
0.10
0.05
3.078 6.314
1.886 2.920
1.638 2.353
1.533 2.132
1.476 2.015
1.440 1.943
1.415 1.895
right tailed area ( α )
0.025
0.01
0.005
12.706 31.821 63.657
4.303
6.965
9.925
3.182
4.541
5.841
2.776
3.747
4.604
2.571
3.365
4.032
2.447
3.143
3.707
2.365
2.998
3.499
0.0025
127.321
14.089
7.453
5.598
4.773
4.317
4.029
0.0010
318.309
22.327
10.215
7.173
5.893
5.208
4.785
0.0005
636.619
31.599
12.924
8.610
6.869
5.959
5.408
Therefore, the t-value with sample size of 7 and has 𝛼 = 0.05 is 1.943.
2. Since the degree of freedom and 𝛼 are already given locate the t-value
on the t-table.
Therefore the t-value is 2.528.
8
3. a. The 95th percentile is the number where 95% of the values lie below it
and 5% lie above it, so you want the right-tail area to be 0.05. Move across
the row, find the column for 0.05, and then locate the t-value using n=30 or
df = 29.
So therefore, the t-value of P95 with n= 30 is 1.699.
b. The 90th percentile is the number where 90% of the values lie below it
and 10% lie above it, so you want the right-tail area to be 0.01. Move across
the row, find the column for 0.05, and then locate the t-value using n=20 or
df = 19.
9
Therefore, the t-value of 90th percentile with n= 20 is 1.328
What’s More
Complete the table below by identifying the degree of freedom, right-tailed
area and the t-value for each percentiles and the sample size.
Percentile
n
1. 60th percentile
2
2. 75th percentile
5
3. 25th percentile
15
4. 90th percentile
7
5. 95th percentile
20
df
10
Right tailed
area
t
What I Have Learned
With steps closer on closing this module, answer the following questions with
at most 3 sentences. Write your answer on a separate sheet of paper.
1. What are the characteristics of a t-distribution? (Give at least 3
characteristics).
2. Enumerate the steps on identifying the t-value using the number of samples
and the right tailed area.
3. How to identify the t-value of a certain percentile using the t-table?
11
What I Can Do
ACTIVITY 1:
Triple Matching Type: Match column A (the t-value) to Column b (the degree of
freedom) and then to column C (the percentile).
A
B
1. 0.277
2. 1.734
3. 1.000
4. 1.345
5. 2.447
C
A. 1
a. 97.5th percentile
B. 6
b. 90th percentile
C. 18
c. 75th percentile
D. 3
d. 95th percentile
E. 14
e. 60th percentile
ACTIVITY 2:
Rowena wants to test a randomly selected 20 students for her study.
Identify the t-value of the following percentiles.
1. 75th percentile
2. 5th percentile
3. 60th percentile
4. 40th percentile
5. 90th percentile
Assessment
A. Multiple Choice. Choose the letter of the best answer. Write the chosen
letter on a separate sheet of paper.
1. Which of the following is NOT a characteristics of t-distribution?
A. Like the normal distribution, the t-distribution has a smooth shape.
B. Like the normal distribution, the t-distribution is symmetric. If you
think about folding it in half at the mean, each side will be the same.
C. Like a standard normal distribution (or z-distribution), the tdistribution has a mean of one.
D. The normal distribution assumes that the population standard
deviation is known. The t-distribution does not make this assumption.
12
2. This refers to the measure of position with data divided into 100 parts.
A. Percentiles
B. Quartiles
C. Deciles
D. Median
3. This refers to the maximum number of logically independent values, which
vary in the data sample.
A. Level of significance
B. Percentiles
C. Degree of freedom
D. Probability
4. What is the right-tailed area if the confidence interval is 75%?
A. 0.05
B. 0.10
C. 0.005
D. 0.25
5. What will be the degree of freedom of the data whose sample size is 18?
A. 17
B. 18
C. 19
D. 20
For number 6 to 8, use the given data below to answer the following questions.
There are 9 students who will be taking up the test, the researcher sets the level of
significance to 0.05. The following are the scores of the nine students:
23, 25, 25, 26, 27, 28, 30, 40, 45
6. What will be the t-value of the given problem?
A. 1.729
B. 1.456
C. 1.612
D. 1.551
7. What will be the value of 25th percentile?
A. 25
B. 26
C. 27
D. 28
13
8. Using the t-table, give the t-value of 95th percentile?
A. 1.729
B. 1.456
C. 1.612
D. 1.551
9. What will be the degree of freedom whose t-value is 0.689?
A. 1
B. 2
C. 3
D. 4
10. Find the t-value of 2.5th percentile using the t-table with a degree of freedom
of 6.
A. 2.570
B. 4.030
C. -2.570
D. -4.030
II. For numbers 11-15, fill out the table below.
Percentile
40th percentile
75th percentile
90th percentile
95th percentile
97.5th percentile
n
2
10
15
20
25
t
Additional Activities
A researcher conducts a diagnostic examination to 20 students. The following are
the interpretation of the diagnostic examination.
1. James got a score which is 60% higher than those who took the diagnostic test.
2. Rowena has a percentile rank of 90.
3. Jimuel got a score which is only 5% higher than those who took the diagnostic
test.
4. Joanna has a percentile rank of 75.
5. Lester is 90% lower than those who took the diagnostic test.
14
, identify the t-value of the percentiles of the following by filling up the table:
Student
1. Joanna
2. James
3. Jimuel
4. Lester
5. Rowena
t-value
Question:
1. Who do you think got the highest score?
2. Who do you think passed the diagnostic examination?
15
What’s New
16
What I Can Do
What's More
A.
1. D e
2. C d
3. A c
4. E b
5. B a
B.
1. 0.688
2. -1.729
3. 0.257
4. -0.257
5. 0.328
What I Know
A.
1. D
2. C
3. A
4. B
5. C
What’s In
A.
6. A
7. A
8. A
9. A
1. 40
2. 50
3. 56
4. 65
5. 90
10.C
B.
B.
1. 75
2. 80
3. 45
4. 75
5. 50%
P
P40
P75
P90
P95
P97.5
n
23
20
25
15
10
t
0.256
0.688
1.318
1.761
2.262
Answer Key
17
Assessment
A.
1. C
2. B
3. C
4. D
5. A
6. A
7. A
8. C
9. C
10. C
B.
1. -0.325
2. 0.703
3. 1.345
4. 1.729
5. 2.064
Additional Activities
1. 0.688
2. 0.257
3. -1.729
4. -1.328
5. 1.328
Questions :
1. Rowena
2. Cannot be
determined
References
Calaca.N, Uy, C. ,Noble, N.M.& Manalo. R. A. (2016). Statistics and Probability. VIBAL
BOOKSTORE.
https://www.dummies.com/education/math/statistics/how-to-find-percentiles-fora-t-distribution/courses.lumenlearning.com.
https://www.jbstatistics.com/r-basics-finding-percentiles-and-areas-for-the-tdistribution/
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For inquiries or feedback, please write or call:
Department of Education – Region III,
Schools Division of Bataan - Curriculum Implementation Division
Learning Resources Management and Development Section (LRMDS)
Provincial Capitol Compound, Balanga City, Bataan
Telefax: (047) 237-2102
Email Address: bataan@deped.gov.ph