So let's again go back to chapter number one. Last
time we discussed chapter one, production and data
collection. And I think we described why learning
statistics distinguish between some of these
topics. And also we explained in details types of
statistics and we mentioned that statistics mainly
has two types either descriptive statistics which
means collecting summarizing and obtaining data
and other type of statistics is called inferential
statistics or statistical inference and this type
of statistics we can draw drawing conclusions and
making decision concerning a population based only
on a sample. That means we have a sample and
sample is just a subset of the population or the
portion of the population and we use the data from
that sample to make some conclusion about the
entire population. This type of statistic is
called inferential statistics. Later, Inshallah,
we'll talk in details about inferential statistics
that will start in Chapter 7. Also, we gave some
definitions for variables, data, and we
distinguished between population and sample. And
we know that the population consists of all items
or individuals about which you want to draw a
conclusion. But in some cases, it's very hard to
talk about the population or the entire
population, so we can select a sample. A sample is
just a portion or subset of the entire population.
So we know now the definition of population and
sample. The other two types, parameter and
statistics. Parameter is a numerical measure that
describes characteristics of a population, while
on the other hand, a sample, a statistic is just
numerical measures that describe characteristic of
a sample. So parameter is computed from the
population while statistic is computed from the
sample. I think we stopped at this point. Why
collect data? I mean what are the reasons for
One of these reasons, for example, a marketing
research analyst needs to assess the effectiveness
of a new television advertisement. For example,
suppose you are a manager and you want to increase
your salaries or your sales. Now, sales may be
affected by advertising. So I mean, if you spend
more on advertising, it means your sales becomes
larger and larger. So you want to know if this
variable, I mean if advertisement is an effective
variable that maybe increase your sales. So that's
one of the reasons why we use data. The other one,
for example, pharmaceutical manufacturers needs to
determine whether a new drug is more effective
than those currently used. For example, for a
headache, we use drug A. Now, a new drug is
produced and you want to see if this new drug is
more effective than drug A that I mean if headache
suppose for example is removed after three days by
using drug A now the question is does B is more
effective it means it reduces your headache in
fewer than three days I mean maybe in two days
That means a drug B is more effective than a drug
A. So we want to know the difference between these
two drugs. I mean, we have two samples. Some
people used drug A and the other used drug B. And
we want to see if there is a significant
difference between the times that is used to
reduce the headache. So that's one of the reasons
why we use statistics. Sometimes an operation
manager wants to monitor manufacturing process to
find out whether the quality of a product being
manufactured is conforming to a company's
standards. Do you know what the meaning of
company's standards?
The regulations of the firm itself. Another
example, suppose here in the school last year, we
teach statistics by using method A. traditional
method. This year we developed a new method for
teaching and our goal is to see if the new method
is better than method A which was used in last
year. So we want to see if there is a big
difference between scores or the average scores
last year and this year. The same you can do for
your weight. Suppose there are 20 students in this
class and their weights are high. And our goal is
to reduce their weights. Suppose they
have a regime or diet for three months or
exercise, whatever it is, then after three months,
we have new weights for these persons. And we want
to see if the diet is effective. I mean, if the
average weight was greater than or smaller than
before diet. Is it clear? So there are many, many
reasons behind using statistics and collecting
data. Now, what are the sources of data? Since
statistics mainly, first step, we have to collect
data. Now, what are the sources of the data?
Generally speaking, there are two sources. One is
called The primary sources and the others
secondary sources. What do you think is the
difference between these two? I mean, what's the
difference between primary and secondary sources?
The primary source is the collector of the data.
He is the analyzer. He analyzes it. And then the
secondary, who collects the data, isn't there.
That's correct. So the primary sources means the
researcher by himself. He should collect the data,
then he can use the data to do his analysis.
That's for the primary. Now, the primary could be
data from political survey. You can distribute
questionnaire, for example, data collected from an
experiment. I mean maybe control or experimental
groups. We have two groups, maybe healthy people
and patient people. So that's experimental group.
Or observed data. That's the primary sources.
Secondary sources, the person performing data
analysis is not the data collector. So he obtained
the data from other sources. For example, it could
be analyzing census data or for example, examining
data from print journals or data published on the
internet. So maybe he goes to the Ministry of
Education and he can get some data. So the data is
already there and he just used the data to do some
analysis. So that's the difference between a
primary and secondary sources. So primary, the
researcher himself, should collect the data by
using one of the tools, either survey,
questionnaire, experiment, and so on. But
secondary, you can use the data that is published
in the internet, for example, in the books, in
governments and NGOs and so on. So these are the
two sources of data. Sources of data fall into
four categories. Number one, data distributed by
an organization or an individual. So that's
secondary source. A design experiment is primary
because you have to design the experiment, a
survey. It's also primary. An observational study
is also a primary source. So you have to
distinguish between a primary and secondary
sources. Any question? Comments? Next.
We'll talk a little bit about types of variables.
In general, there are two types of variables. One
is called categorical variables or qualitative
variables, and the other one is called numerical
or quantitative variables. Now, for example, if I
ask you, what's
your favorite color? You may say white, black,
red, and so on. What's your marital status? Maybe
married or unmarried, and so on. Gender, male,
either male or female, and so on. So this type of
variable is called qualitative variables. So
qualitative variables have values that can only be
placed into categories, such as, for example, yes
or no. For example, do you like orange?
The answer is either yes or no. Do you like
candidate A, for example, whatever his party is?
Do you like it? Either yes or no, and so on. As I
mentioned before, gender, marital status, race,
religions, these are examples of qualitative or
categorical variables. The other type of variable
which is commonly used is called numerical or
quantitative data. Quantitative variables have
values that represent quantities. For example, if
I ask you, what's your age? My age is 20 years old
or 18 years old. What's your weight? Income.
Height? Temperature? Income. So it's a number.
Weight, maybe my weight is 70 kilograms. So
weight, age, height, salary, income, number of
students, number of phone calls you received on
your cell phone during one hour, number of
accidents happened in street and so on. So that's
the difference between numerical variables and
qualitative variables.
Anyone of you just give me one example of
qualitative and quantitative variables. Another
examples. Just give me one example for qualitative
data.
Qualitative or quantitative.
Political party, either party A or party B. So
suppose there are two parties, so I like party A,
she likes party B and so on. So party in this case
is qualitative variable, another one.
So types of courses, maybe business, economics,
administration, and so on. So types of courses.
Another example for quantitative variable or
numerical variables.
So production is a numerical variable. Another
example for quantitative.
Is that produced by this company? Number of cell
phones, maybe 20, 17, and so on. Any question?
Next. So generally speaking, types of data, data
has two types, categorical and numerical data. As
we mentioned, marital status, political party, eye
color, and so on. These are examples of
categorical variables. On the other hand, a
numerical variable can be split or divided into
two parts. One is called discrete and the other is
continuous, and we have to distinguish between
these two. For example, Number of students in this
class, you can say there are 60 or 50 students in
this class. You cannot say there are 50.5
students. So number of students is discrete
because it takes only integers. While for
continuous type of numerical variables, you can
say that my weight is 80.5 kilograms. so it makes
sense that your weight is not exactly 80 kilograms
it might be 80.6 or 80.5 and so on so discrete
takes only integers while continuous takes any
value I mean any real number so that's the
difference between discrete and continuous number
of phone or number of calls you have received this
morning, maybe one, zero, nine, and so on,
discrete. Number of patients in the hospital,
discrete, and so on. But when we are talking about
income, maybe my income is 1,000.5 shekel. It
could be. It's continuous because my income can be
any number between, for example, 1,000 and 10,000.
It takes any value in this interval from 1,000 to
10,000. So it types of continuous rather than our
continuous variable. So that's the two types of
data, categorical and numerical. And numerical
also has two types, either discrete or continuous.
Later in Chapter 6, we'll talk more details about
one of the most distribution statistics, for
continuous, one which is called normal
distribution. That will be later, inshallah. As we
mentioned last time, at the end of each chapter,
there is a section or sections, sometimes there
are two sections, talks about computer programs.
How can we use computer programs in order to
analyze the data? And as we mentioned last time,
you should take a course on that. It's called
Computer and Data Analysis or SPSS course. So we
are going to skip the computer programs used for
any chapters in this book. And that's the end of
chapter number three. Any questions?
Let's move. quickly on chapter three.
Chapter three maybe is the easiest chapter in this
book. It's straightforward. We have some formulas
to compute some statistical measures. And we
should know how can we calculate these measures
and what are the meaning of your results. So
chapter three talks about numerical descriptive
measures. In this chapter, you will learn, number
one, describe the probabilities of central
tendency, variation, and shape in numerical data.
In this lecture, we'll talk in more details about
some of the center tendency measures. Later, we'll
talk about the variation, or spread, or
dispersion, and the shape in numerical data. So
that's part number one. We have to know something
about center tendency, variation, and the shape of
the data we have. to calculate descriptive summary
measures for a population. So we have to calculate
these measures for the sample. And if we have the
entire population, we can compute these measures
also for that population. Then I will introduce in
more details about something called Paxiplot. How
can we construct and interpret a Paxiplot? That's,
inshallah, next time on Tuesday. Finally, we'll
see how can we calculate the covariance and
coefficient of variation and coefficient, I'm
sorry, coefficient of correlation. This topic
we'll introduce in more details in chapter 11
later on. So just I will give some brief
notation about coefficient of correlation, how can
we compute the correlation coefficient? What's the
meaning of your result? And later in chapter 11,
we'll talk in more details about correlation and
regression. So these are the objectives of this
chapter. There are some basic definitions before
we start. One is called central tendency. What do
you mean by central tendency? Central tendency is
the extent to which all data value group around a
typical or numerical or central value. So we are
looking for a point that in the center, I mean,
the data points are gathered or collected around a
middle point, and that middle point is called the
central tendency. And the question is, how can we
measure that value? We'll talk in details about
mean, median, and mode in a few minutes. So the
central tendency, in this case, the data values
grouped around a typical or central value. Is it
clear? So we have data set, large data set. Then
these points are gathered or grouped around a
middle point, and this point is called central
tendency, and it can be measured by using mean,
which is the most common one, median and the moon.
Next is the variation, which is the amount of
dispersion. Variation is the amount of dispersion
or scattering of values. And we'll use, for
example, range, variance or standard deviation in
order to compute the variation. Finally, We have
data, and my question is, what's the shape of the
data? So the shape is the pattern of distribution
of values from the lowest value to the highest. So
that's the three definitions we need to know
before we start. So we'll start with the easiest
one, measures of central tendency. As I mentioned,
there are three measures. median and moon. And our
goal or we have two goals actually. We have to
know how to compute these measures. Number two,
which one is better? The mean or the median or the
moon?
So the mean sometimes called the arithmetic mean.
Or in general, just say the mean. So often we use
the mean. And the mean is just sum
of the values divided by the sample size. So it's
straightforward. We have, for example, three data
points. And your goal is to find the average or
the mean of these points. They mean it's just some
of these values divided by the sample size. So for
example, if we have a data X1, X2, X3 up to Xn. So
the average is denoted by X bar. This one is
pronounced as X bar and X bar is just sum of Xi.
It is summation, you know this symbol, summation
of sigma, summation of Xi and I goes from one to
N. divided by N which is the total number of
observations or the sample size. So it means X1
plus X2 all the way up to XN divided by N gives
the mean or the arithmetic mean. So X bar is the
average which is the sum of values divided by the
number of observations. So that's the first
definition. For example,
So again, the mean is the most common measure of
center tendency. Number two, the definition of the
mean. Sum of values divided by the number of
values. That means the mean takes all the values,
then divided by N. it makes sense that the mean is
affected by extreme values or outliers. I mean, if
the data has outliers or extreme values, I mean by
extreme values, large or very, very large values
and small, small values. Large values or small
values are extreme values. Since the mean takes
all these values and sums all together, doesn't
divide by n, that means The mean is affected by
outliers or by extreme values. For example,
imagine we have simple data as 1, 2, 3, 4, and 5.
Simple example. Now, what's the mean? The mean is
just add these values, then divide by the total
number of observations. In this case, the sum of
these is 15. N is five because there are five
observations. So X bar is 15 divided by 5, which
is 3. So straightforward. Now imagine instead of
5, this number 5, we have a 10. Now 10, there is a
gap between 4, which is the second largest, and
the maximum, which is 10. Now if we add these
values, 1, 2, 3, 4, and 10, then divide by 5, the
mean will be 4. If you see here, we just added one
value, or I mean, we replaced five by 10, and the
mean changed dramatically from three to four.
There is big change between three and four, around
25% more. So that means outliers or extreme values
affected the mean. So take this information in
your mind because later we'll talk a little bit
about another one. So the mean is affected by
extreme values. Imagine another example. Suppose
we have data from 1 to 9. 1, 2, 3, 4, 6, 7, 8, 9.
Now the mean of these values, some divide by n. If
you sum 1 through 9, summation is 45. Divide by 9,
which is 5. So the sum of these values divided by
N gives the average, so the average is 5. Now
suppose we add 100 to the end of this data. So the
sum will be 145 divided by 10, that's 14.5. Now
the mean was 5. Then after we added 100, it
becomes 14.5. Imagine the mean was 5, it changed
to 14.5. It means around three times. So that
means outliers affect the mean much more than the
other one. We'll talk a little later about it,
which is the median. So keep in mind outliers
affected the mean in this case. Any question? Is
it clear? Yes. So, one more time. The mean is
affected by extreme values. So that's for the
mean. The other measure of center tendency is
called the median. Now, what's the median? What's
the definition of the median from your previous
studies? What's the median? I mean, what's the
definition of the median? Now the middle value,
that's correct, but after we arrange the data from
smallest to largest or largest to smallest, so we
should arrange the data, then we can figure out
the median. So the median is the middle point, but
after we arrange the data from smallest to largest
or vice versa. So that's the definition of the
median. So in an ordered array, so we have to have
order array, the median is the middle number. The
middle number means 50 percent of the data below
and 50 percent above the median because it's
called the median, the value in the middle after
you arrange the data from smallest to largest.
Suppose I again go back to the previous example
when we have data 1, 2, 3, 4, and 5. Now for this
specific example as we did before, now the data is
already ordered. The value in the middle is 3
because there are two pillows.
And also there are the same number of observations
above it. So 3 is the median. Now again imagine we
replace 5, which is the maximum value, by another
one which is extreme one, for example 10. In this
case, the median is still 3. Because the median is
just the value of the middle after you arrange the
data. So it doesn't matter what is the highest or
the maximum value is, the median in this case is
three. It doesn't change. That means the median is
not affected by extreme values. Or to be more
precise, we can say that The median is affected by
outlier, but not the same as the mean. So affect
the mean much more than the median. I mean, you
cannot say for this example, yes, the median is
not affected because the median was three, it
becomes three. But in another examples, there is
small difference between all.
Extreme values affected the mean much more than
the median. If the dataset has extreme values, we
have to use, what do you think, the mean or the
median? The median. So in case or in the presence
of extreme values or outliers, we have to use the
median, not the mean. But in general, we use If
the data is free of outliers, I mean if the data
has not extreme values, then you can use the mean.
The mean is much better than the median in this
case. But if the data has extreme values or
outliers, we should use the median instead of the
mean. Any question? So these are the most common
center tendency measures in statistics, the mean
and the median. And keep in mind, your data should
be numeric. I mean, you cannot use the mean or the
median for qualitative or categorical data, for
example, gender, males or females. You cannot say
the mean of gender or sex is 1.5. It doesn't make
sense. It should be numerical data to use the mean
or the median. So the mean and the median is used
only for numerical data. And we have to
distinguish between mean and median. Mean is used
for data that has not outliers or extreme values,
while the median is used for data that has
outliers or extreme values. Sometimes better to
report both. I mean, sometimes better to report
mean and the median. So you just say the sales for
this company is, for example, $500,000. And the
median, for example, is 550,000. You can see that.
Is it clear?
If you have a small data, it's straightforward and
it's very easy to locate the median. But in case
of large dataset, how can we locate the median?
It's not easy. Just look at the data and you can
say this is the median. It's not easy task. So we
need a rule that locate the median. The location
of the median when the values are in numerical
order from smallest to largest is N plus one
divided by two. That's the position of the median.
If we go back a little bit to the previous
example, here N was five. So the location was
number three, because n plus one divided by two,
five plus one divided by two is three. So location
number three is the median. Location number one is
one, in this case, then two, then three. So
location number three is three. But maybe this
number is not three, and other value maybe 3.1 or
3.2. But the location is number three. Is it
clear? So that's the location. If it is odd, you
mean by odd number, five, seven and so on. So if
the number of values is odd, the median is the
middle number. Now let's imagine if we have even
number of observations. For example, we have one,
two, three, four, five and six. So imagine numbers
from one up to six. What's the median? Now three
is not the median because there are two
observations below three. And three above it. And
four is not the median because three observations
below, two above. So three and four is the middle
value. So just take the average of two middle
points, And that will be the median. So if n is
even, you have to locate two middle points. For
example, n over 2, in this case, we have six
observations. So divide by 2, not n plus 1 divided
by 2, just n over 2. So n over 2 is 3. So place
number 3, and the next one, place number 4, these
are the two middle points. Take the average of
these values, then that's your median. So if N is
even, you have to be careful. You have to find two
middle points and just take the average of these
two. So if N is even, the median is the average of
the two middle numbers. Keep in mind, when we are
saying N plus 2, N plus 2 is just the position of
the median, not the value, location. Not the
value. Is it clear? Any question? So location is
not the value. Location is just the place or the
position of the medium. If N is odd, the position
is N plus one divided by two. If N is even, the
positions of the two middle points are N over two
and the next term or the next point.
Last measure of center tendency is called the
mood.
The definition of the mood, the mood is the most
frequent value. So sometimes the mood exists,
sometimes the mood does not exist. Or sometimes
there is only one mood, in other cases maybe there
are several moods. So a value that occurs most
often is called the mood. The mood is not affected
by extreme values. It can be used for either
numerical or categorical data. And that's the
difference between mean and median and the mood.
Mean and median is used just for numerical data.
Here, the mood can be used for both, categorical
and numerical data. Sometimes, as I mentioned,
there may be no mood or the mood does not exist.
In other cases, there may be several events. So
the mood is the value that has the most frequent.
For example, if you look at this data, one is
repeated once, three is the same one time, five is
repeated twice. seven is one nine is repeated
three times and so on so in this case nine is the
mood because the mood again is the most frequent
value on
the right side there are some values zero one two
three up to six now each one is repeated once so
in this case the mood does not exist I mean there
is no mood So generally speaking, the mood is the
value that you care most often. It can be used for
numerical or categorical data, not affected by
extreme values or outliers. Sometimes there is
only one mood as this example. Sometimes the mood
does not exist. Or sometimes there are several
moods. And so that's the definitions for mean,
median, and the mood. I will give just a numerical
example to know how can we compute these measures.
This data, simple data, just for illustration, we
have house prices. We have five data points, $2
million. This is the price of house A, for
example. House B price is 500,000. The other one
is 300,000. And two houses have the same price as
100,000. Now, just to compute the mean, add these
values or sum these values, which is three
million, divide by number of houses here, there
are five houses, so just three thousand divided by
five, six hundred thousand. The median, the value
in the median, after you arrange the data from
smallest to largest, Or largest smallest. This
data is already arranged from largest smallest or
smallest large. It doesn't matter actually. So the
median is $300,000. Make sense? Because there are
two house prices above and two below. So the
median is $300,000. Now if you look at these two
values, the mean for this data equals 600,000 and
the median is 300,000. The mean is double the
median. Do you think why there is a big difference
in this data between the mean and the median?
Which one? Two million dollars is extreme value,
very large number. I mean, if you compare two
million dollars with the other data sets or other
data values, you will see there is a big
difference between two million and five hundred.
It's four times, plus about three hundred
thousands, around seven times and so on. For this
value, the mean is affected. Exactly. The median
is resistant to outliers. It's affected but little
bit. For this reason, we have to use the median.
So the median makes more sense than using the
mean. The mode is just the most frequent value,
which is 100,000, because this value is repeated
twice. So that's the whole story for central
tendency measures, mean, median, and 1. Now the
question again is which measure to use? The mean
is generally used. The most common center tendency
is the mean. We can use it or we should use it
unless extreme values exist. I mean if the data
set has no outliers or extreme values, we have to
use the mean instead of the median.
The median is often used since the median is not
sensitive to extreme values. I mean, the median is
resistant to outliers. It remains nearly in the
same position if the dataset has outliers. But the
median will be affected either to the right or to
the left tail. So we have to use the median if the
data has extreme values. For example, median home
prices for the previous one may be reported for a
region that is less sensitive to outliers. So the
mean is more sensitive to outliers than the
median. Sometimes, I mean in some situations, it
makes sense to report both the mean and the
median. Just say the mean for this data for home
prices is 600,000 while the median is 300,000. If
you look at these two figures, you can tell that
there exists outlier or the outlier exists because
there is a big difference between the mean and the
median. So that's all for measures of central
tendency. Again, we explained three measures,
arithmetic mean, median, and mode. And arithmetic
mean again is denoted by X bar is pronounced as X
bar and just summation of X divided by N. So
summation Xi, i goes from 1 up to N divided by the
total number of observations. The median, as we
mentioned, is the value in the middle in ordered
array. After you arrange the data from smallest to
largest or vice versa, then the median is the
value in the middle. The mode is the most frequent
observed value. And we have to know that mean and
median are used only for numerical data, while the
mode can be used for both numerical and
categorical data. That's all about measures of
central tendency. Any question?
Let's move to measures of variation. It's another
type of measures. It's called measures of
variation, sometimes called measures of spread.
Now, variation can be computed by using range,
variance, standard deviation, and coefficient of
variation. So we have four types, range, variance,
standard deviation, and coefficient of variation.
Now, measures of variation give information on the
spread. Now, this is the first difference between
central tendency measures and measures of
variation. That one measures the central value or
the value in the middle. Here, it measures the
spread. Or variability. Or dispersion of the data.
Do you know what is dispersion? Dispersion.
Tabaad. So major variation given formation with
the spread. Spread or variation or dispersion of
the data values. Now if you look at these two bell
shapes.
Both have the same center. The center I mean the
value in the middle. So the value in the middle
here for figure
graph number one is the same as the value for the
other graph. So both graphs have the same center.
But if you look at the spread, you will see that
figure A is less spread than figure B. Now if you
look at this one, the spread here, is much less
than the other one. Even they have the same
center, the same mean, but figure A is more spread
than figure B. It means that the variation in A is
much less than the variation in figure B. So it
means that the mean is not sufficient to describe
your data. Because maybe you have two datasets and
both have the same mean, but the spread or the
variation is completely different. Again, maybe we
have two classes of statistics, class A and class
B. The center or the mean or the average is the
same for each one. For example, maybe the average
of this class is 70. The average of class B is
also 70. But the scores are scattered. or spread
out in class A maybe much more than in class B. So
the mean is not sufficient to describe the data.
You have to say that the mean equals such and such
and the spread. And one of these measures we'll
talk later about range and variance standard
deviation. So I mean, The mean by itself is not
sufficient to describe the data. You have to use
something else to measure the variation or the
spread of the data. Make sense?
The first measure of variation, the easiest one,
is called the range. The range is the simplest
measure of variation. The range is just the
difference or the distance between the largest and
the smallest value. For example, suppose the
minimum score for this class is 40 and the maximum
is 90. So the range is 50, 90 minus 40. Now
imagine that the minimum score for this class is
60 and the maximum is 80, so 20. If we replace 80
by 100, I mean the minimum is 60 and the maximum
is 100, it's 40. That means a range is affected by
outliers because it depends only on two values.
maximum and minimum value. So it should be
affected by outliers. So range is sensitive to
outliers. So if the data has the data set has
outliers, then in this case, you have to avoid
using range because range only based on two
values. So it should be affected by outliers. Now
for the for simple example, suppose we have this
data. The minimum value is one. I mean, the
smallest value is one, and the largest or the
maximum is 13. So it makes sense that the range of
the data is the difference between these two
values. So 13 minus one is 12. Now, imagine that
we just replace 13 by 100. So the new range will
be equal to 100 minus 199. So the previous range
was 12. It becomes now 99 after we replace the
maximum by 100. So it means that range is affected
by extreme values. So the mean and range both are
sensitive to outliers. So you have to link between
measures of center tendency and measures of
variation. Mean and range are affected by
outliers. The mean and range are affected by
outliers. This is an example. So it's very easy to
compute the mean. Next, if you look at why the
range can be misleading.
Sometimes you report the range and the range does
not give an appropriate answer or appropriate
result because number
one ignores the way in which the data are
distributed. For example, if you look at this
specific data, we have data seven, eight, nine,
ten, eleven and twelve. So the range is five.
Twelve minus seven is five. Now if you look at the
other data, The smallest value was seven.
And there is a gap between the smallest and the
next smallest value, which is 10. And also we have
12 is repeated three times. Still the range is the
same. Even there is a difference between these two
values, between two sets. we have seven, eight,
nine up to 12. And then the other data, we have
seven, 10, 11, and 12 three times. Still, the
range equals five. So it doesn't make sense to
report the range as a measure of variation.
Because if you look at the distribution for this
data, it's completely different from the other
dataset. Even though it has the same range. So
range is not used in this case. Look at another
example.
We have data. All the data ranges, I mean, starts
from 1 up to 5. So the range is 4. If we just
replace the maximum, which is 5, by 120. So the
range is completely different. the range becomes
119. So that means range
is sensitive to outliers. So we have to avoid
using range in case of outliers or extreme values.
I will stop at the most important one, the
variance, for next time inshallah. Up to this
point, any questions?
Okay, stop at this point if
you have any question.
So later we'll discuss measures of variation and
variance, standard deviation up to the end of this
chapter.
So again, the range is sensitive to outliers. So
we have to avoid using range in this case. And
later we'll talk about the variance, which is the
most common measures of variation for next time,
inshallah.