The document discusses various measures of central tendency and spread that can be used to summarize sample data. It describes the arithmetic mean as the average value found by summing all data points and dividing by the sample size. The mode is defined as the most frequent data point. The median is the middle value when data is arranged in order. The interquartile range is introduced as a measure of spread or dispersion in the data. Formulas are provided for calculating these metrics from both raw and grouped frequency data.
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If you happen to like this powerpoint, you may contact me at flippedchannel@gmail.com
I offer some educational services like:
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When fitting loss data (insurance) to a distribution, often the parameters that provide a good overall fit will understate the density in the tail.
This method allows one to split the distribution into 2 portions, and use a Pareto distribution to fit the tail.
Presented at the CAS Spring Meeting in Seattle, May 2016.
Missing Parts I don’t think you understood the assignment.docxannandleola
Missing Parts:
I don’t think you understood the assignment. I am looking at it, all I see is where you entered
SAS codes and then that’s it. These SAS codes you inputted, I’d like to see some results, such as
these things I am about to mention:
Part I)
1. (2 pts.) Import the data into your software. Be sure to check that your data looks
exactly like the original data before proceeding! 2. (2 pts.) For BOTH of your
original quantitative variables, create TWO categorized versions based upon cutoffs
of your choice. One binary version and one multi-level version with 3-5 groups. Use
numbers for the new variables to represent the groups. No group should have less
than 10% of the overall sample. Be sure you define your groups so that they do not
overlap and you do not miss any observations. • In SPSS this can be done using
TRANSFORM and RECODE INTO DIFFERENT VARIABLE. • In SAS you need
to use a DATA step with IF-THEN statements to create the new variables. 3. (2 pts.)
Create translations which provide the range of values for the variables created in
Question 3. • In SPSS this is done in the variable view using the “Values” column. •
In SAS you need to create the formats using PROC FORMAT and then assign those
formats to the appropriate variables using a DATA step. 4. (3 pts.) Label all
variables with descriptive titles. • In SPSS this is done in the variable view using the
“Label” column. • In SAS you need to use a DATA step which includes a LABEL
statement.
All the codes I’m looking at, I didn’t need to see them, I expect to see them in a table. I’ve
similar exercises, and that’s not how they look.
PART II)
Part 2: Descriptive Summary of Each Variable 5. (6 pts.) Calculate the sample size, sample
mean, sample median, sample standard deviation, min, max, Q1, Q3, and 95% confidence
interval for the population mean for your two quantitative variables. Provide the software
output containing these results in your solution. 6. (6 pts.) Construct a histogram, boxplot,
and QQ-plot for your two quantitative variables. Provide only the graphs in your solution.
7. (8 pts.) Construct a frequency table for each of the four variables created in Question 3.
8. (6 pts.) Provide a brief discussion of the distribution of your two main variables using as
much of the information in Questions 5-7 as possible (and yet remain as concise as
possible).
Where did you do all these calculations; I didn’t see anything. I did see a histogram, that’s all I
saw. Where’s the box plot, QQ plot, there was no graph. Also, you didn’t provide any discussion.
PART III)
Part 3: Case QQ - Using the two quantitative variables 9. (2 pts.) Construct a scatterplot.
Provide only this plot in your solution. 10. (2 pts.) Regardless of whether it is appropriate,
calculate Pearson’s correlation coefficient. Provide the output containing the estimate and
the p-value. 11. (3 pts.) Regardless of whether it is a ...
Application of Machine Learning in AgricultureAman Vasisht
With the growing trend of machine learning, it is needless to say how machine learning can help reap benefits in agriculture. It will be boon for the farmer welfare.
A measure of central tendency (also referred to as measures of centre or central location) is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or centre of its distribution.
Similar to Chapter 9 learning more about sample data(1) (20)
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
1. 2013/05/22
1
STATISTICS
X-Kit Textbook
Chapter 9
Precalculus Textbook
Appendix B: Concepts in Statistics
Par B.2
CONTENT
THE GOAL
Look at ways of summarising a large
amount of sample data in just one or two
key numbers.
Two important aspects of a set of data:
•The LOCATION
•The SPREAD
MEASURES OF CENTRAL TENDENCY
(LOCATION)
Arithmetic Mean (Average)
Mode (the highest point/frequency)
Median (the middle observation)
Number of fraudulent cheques received at a
bank each week for 30 weeks
Week
1
2 3 4 5 6 7 8 9 10
5 3 8 3 3 1 10 4 6 8
Week
11
12 13 14 15 16 17 18 19 20
3 5 4 7 6 6 9 3 4 5
Week
21
22 23 24 25 26 27 28 29 30
7 9 4 5 8 6 4 4 10 4
ARITHMETIC MEAN
• 𝒙 =
𝟏𝟔𝟒
𝟑𝟎
= 𝟓. 𝟒𝟕
• To calculate the MEAN add all the data points
in our sample and divide by die number of
data points (sample size).
• The MEAN can be a value that doesn’t
actually match any observation.
• The MEAN gives us useful information about
the location of our frequency distribution.
2. 2013/05/22
2
GRAPH
0
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8 9 10
Frequency
Frequency
CALCULATE THE MEAN
Raw Data
• 𝑥 =
𝑥
𝑛
• 𝑥 is data
points
• 𝑛 is number
of
observations
Frequency
Table
• 𝑥 =
𝑥𝑓
𝑛
• 𝑥 is data
points
• 𝑛 is number
of
observations
• 𝑓 is the
frequency
Frequency
Table (Intervals)
• 𝑥 =
𝑥𝑓
𝑛
• 𝑥 is midpoints
for intervals
• 𝑛 is number
of
observations
• 𝑓 is the
frequency
CALCULATE THE MEAN - FREQUENCY TABLE:
NUBEROFFRAUDULENT CHEQUESPERWEEK
Distinct Values TallyMarks Frequency
1 / 1
2 0
3 //// 5
4 //// // 7
5 //// 4
6 //// 4
7 // 2
8 /// 3
9 // 2
10 // 2
Truck Data: weights (in tonnes) of 20 fully
loaded trucks
Truck
1
2 3 4 5 6 7 8 9 10
Weight
4.54
3.81 4.29 5.16 2.51 4.63 4.75 3.98 5.04 2.80
Truck
11
12 13 14 15 16 17 18 19 20
Weight
2.52
5.88 2.95 3.59 3.87 4.17 3.30 5.48 4.26 3.53
CALCULATE THE MEAN - GROUPED
FREQUENCY TABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
Class Intervals Frequency Midpoint
𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 𝟐. 𝟓 + 𝟑. 𝟎 ÷ 𝟐 = 2.75
𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 3.25
𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 3.75
𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 4.25
𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 4.75
𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 5.25
𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 5.75
MODE
•The mode is the interval with the
HIGHEST FREQUENCY.
•There can be two or more modes in a set
of data – then the mode would not be a
good measure of central tendency.
•MULTI-MODAL data consist of more than
one mode.
•UNI-MODAL data consist of only one
mode.
4. 2013/05/22
4
DON’T FALL INTO THE COMMON TRAP
• The median is NOT the middle of the range of
observations, for example
1, 1, 1, 1, 1, 3, 9
The median is 1 (the middle observation).
The middle of the range (9 – 1) is 5! Big
difference!
MEDIAN
Odd Number of
Observations,
for example 7
Median Position
𝒏+𝟏
𝟐
Even Number of
Observations,
for example30
Median Position
half-way between
𝒏
𝟐
𝒂𝒏𝒅 (
𝒏
𝟐
+ 𝟏)
FINDTHE MEDIAN -FREQUENCYTABLE:
NUBER OF FRAUDULENT CHEQUES PERWEEK
Distinct Values Frequency Cumulative
Frequency
1 1 1
2 0 1
3 5 6
4 7 13
5 4 17
6 4 21
7 2 23
8 3 26
9 2 28
10 2 30
FIND THE MEDIAN - GROUPED FREQUENCY
TABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
ClassIntervals Frequency Midpoint
𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 𝟐. 𝟓 + 𝟑. 𝟎 ÷ 𝟐 = 2.75
𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 3.25
𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 3.75
𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 4.25
𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 4.75
𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 5.25
𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 5.75
FIND THE MEDIAN FROM A GROUPED
FREQUENCY TABLE
•Median (middle observation)?
•Find the class interval in which that
observation lies.
?
CALCULATIONS
Raw Data
Mean
Mode
Median
Frequency Table
(Ungrouped
Data)
Mean
Mode
Median
Frequency Table
(Grouped Data)
Mean
Mode
Median
5. 2013/05/22
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HOW TO CHOOSE THE BEST MEASURE OF
LOCATION?
• When choosing the best measure of location, we
need to look as the SHAPE of the distribution.
• For nearly symmetric data, the mean is the best
choice.
• For very skewed (asymmetric) data, the mode or
median is better.
• The mean moves further along the tail than the
median, it is more sensitive to the values far from
the centre.
SYMMETRIC histogram:
Mean = Median = Mode
A POSITIVELY SKEWED (skewed to the right)
histogram has a longer tail on the right side:
Mode < Median < Mean
A NEGATIVELY SKEWED (skewed to the left)
histogram has a longer tail on the left side:
Mean < Median < Mode
PROBLEM
•We can find two very different data sets (one
distribution very spread out and another very
concentrated) with measures of central
tendency EQUAL.
•To find a true idea of our sample, we have to
MEASURE THE SPREAD OF A DISTRIBUTION,
called the spread dispersion.
MEASURESOF SPREAD(DISPERSION)
Interquartile Range
Variance
Standard Deviation
6. 2013/05/22
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MEASURINGSPREAD
•Think of a distribution in terms of
percentages, a horizontal axis equally divided
into 100 percentiles.
•The 10th percentile marks the point below
which 10% of the observations fall, and
above which 90% of observations fall.
•The 50th percentile, below which 50% of the
observations lie, is the median.
WORKINGWITH A PERCENTILE
• 𝑝% of the observationfall belowthe 𝑝 𝑡ℎ percentile.
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 =
𝒑
𝟏𝟎𝟎
𝒏 + 𝟏
• Workingwith the example on fraudulentcheques:
1, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6,
7, 7, 8, 8, 8, 9, 9, 10, 10
𝑷 𝟓𝟎 =
𝟓𝟎
𝟏𝟎𝟎
𝟑𝟎 + 𝟏 = 𝟏𝟓. 𝟓
• 15.5 tells us where to find our 50th percentile.
• 15 tells us which observation to go to, and 0.5 tells us how far to
move along the space between that observation and the next
highest one.
FORMULA
• 𝑷 𝟓𝟎 = 𝒙 𝟏𝟓 + 𝟎. 𝟓 𝒙 𝟏𝟔 − 𝒙 𝟏𝟓
𝑷 𝒑 = 𝒙 𝒌 + 𝒅 𝒙 𝒌+𝟏 − 𝒙 𝒌
• 𝑃 means percentile
• 𝑝 tell us which percentile
• 𝑘 the whole number calculated from the
position
• 𝑑 the decimal fraction calculated from the
position
WORKINGWITH PERCENTILESFROMUNGROUPEDFREQUENCYDATA:
NUBEROFFRAUDULENT CHEQUESPERWEEK
Distinct Values Frequency Cumulative Frequency
1 1 1
2 0 0 + 1 = 1
3 5 1 + 5 = 6
4 7 6 + 7 = 13
5 4 13 + 4 = 17
6 4 17 + 4 = 21
7 2 21 + 2 = 23
8 3 23 + 3 = 26
9 2 26 + 2 = 28
10 2 28 + 2 = 30
WORKING WITH PERCENTILES (AND
MEDIAN) FROM GROUPED DATA
• To identify the class interval 𝑳 < 𝒙 ≤ 𝑼 containing the
𝑝 𝑡ℎ percentile:
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 =
𝒑
𝟏𝟎𝟎
𝒏 + 𝟏
• The decimal fraction for grouped data is:
𝒅 =
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏−𝑺𝒖𝒎 𝒐𝒇 𝒄𝒍𝒂𝒔𝒔 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒊𝒆𝒔 𝒕𝒐 𝑳
𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝒄𝒍𝒂𝒔𝒔 𝑳 < 𝒙 ≤ 𝑼
• Calculate the 𝑝 𝑡ℎ percentile:
𝑷 𝒑 ≈ 𝑳 + 𝒅 𝑼 − 𝑳
FIND THE MEDIAN - GROUPED FREQUENCY
TABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
Class Intervals Frequency CumulativeFrequency
𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 4
𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 5
𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 10
𝟒. 𝟎 < 𝐱 ≤ 𝟒. 𝟓 3 13
𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 16
𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 19
𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 20
7. 2013/05/22
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FIND THEMEDIAN-GROUPEDFREQUENCYTABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
• To identify the class interval 𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 containing
the 50 𝑡ℎ percentile:
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 =
𝟓𝟎
𝟏𝟎𝟎
𝟐𝟎 + 𝟏 = 𝟏𝟎. 𝟓
• The decimal fraction for grouped data is:
𝒅 =
𝟏𝟎.𝟓 − 𝟏𝟎
𝟑
=
𝟏
𝟔
• Calculate the 𝑝 𝑡ℎ percentile:
𝑷 𝟓𝟎 ≈ 𝟒. 𝟎 + 𝒅 𝟒. 𝟓 − 𝟒. 𝟎 = 𝟒. 𝟎𝟖𝟑𝟑𝟑
MEASURINGSPREAD
• If we measure the DIFFERENCE in value between
one percentile and another, this would give us an
idea of how widely our data is spread out.
• INTERQUARTILE RANGE (IQR) = 75th – 25th Percentiles
• The bigger the IQR, the more spread out the data.
• The 75th percentile ≥ 25th percentile, therefor the
IQR ≥ 0 .
• We tend to use the MEDIAN (as measure of
central tendency) together with the IQR.
FIND THE IQR - GROUPED FREQUENCY
TABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
ClassIntervals Frequency CumulativeFrequency
𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 4
𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 5
𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 10
𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 13
𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 16
𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 19
𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 20
FIND THEMEDIAN-GROUPEDFREQUENCYTABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
• To identify the class interval 𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 containing
the 75 𝑡ℎ percentile:
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 =
𝟕𝟓
𝟏𝟎𝟎
𝟐𝟎 + 𝟏 = 𝟏𝟓. 𝟕𝟓
• The decimal fraction for grouped data is:
𝒅 =
𝟏𝟓. 𝟕𝟓 − 𝟏𝟑
𝟑
= 𝟎. 𝟗𝟏𝟕
• Calculate the 𝑝 𝑡ℎ percentile:
𝑷 𝟕𝟓 ≈ 𝟒. 𝟓 + 𝒅 𝟓. 𝟎 − 𝟒. 𝟓 = 𝟒. 𝟗𝟓𝟖
FIND THEMEDIAN-GROUPEDFREQUENCYTABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
• To identify the class interval 𝟑. 𝟓 < 𝒙 ≤ 𝟒.0 containing
the 25 𝑡ℎ percentile:
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 =
𝟐𝟓
𝟏𝟎𝟎
𝟐𝟎 + 𝟏 = 𝟓. 𝟐𝟓
• The decimal fraction for grouped data is:
𝒅 =
𝟓. 𝟐𝟓 − 𝟓
𝟓
= 𝟎. 𝟎𝟓
• Calculate the 𝑝 𝑡ℎ percentile:
𝑷 𝟐𝟓 ≈ 𝟑. 𝟓 + 𝒅 𝟒. 𝟎 − 𝟑. 𝟓 = 𝟑. 𝟓𝟐𝟓
• IQR = 4.958 – 3.525 = 1.433
MEASURINGSPREAD
• When we use the MEAN as our measure of central
tendency, we usually choose A MEASURE OF HOW FAR
THE DATA IS SPREAD OUT AROUND THE MEAN.
• Two measures of spread that are based on the mean are
the VARIANCE and the STANDARD DEVIATION.
• An advantage of standard deviation is that it is measured
in the same units as the original observations.
• The variance and standard deviation are closely related.
• The variance (𝒔 𝟐 or 𝝈 𝟐) is the square of the standard
deviation (𝒔 or 𝝈).