Measures of Position

In today's chapter we will see what are the measures of position? In this chapter we will be covering three topics.

First, what are quartiles, second what is percentiles and third, what is the interquartile range? So, firstly, we will see what are quartiles.

A quartile defines a particular part of a data set.

It means that suppose that I have one central limit, okay, I have any limit below which and above it, in that distribution the data lies.

How much is my data divided between left and right that is defined by my quartiles.

So, basically quartile divides my data into four parts, which is basically equal.

So, I get three quarters because of it, Q1, Q2 and Q3.

We call a Q1 quartile as the 1^st quartile, Q2 as

the second quartile and Q3 as the third quartile.

What is its significance? Let's see it once.

If I say that Q1 is my that percentile in which 25% of my data lies below the distribution and 75% lies above the distribution.

To understand it easily.

Let's take the example of Q2 which is my second quartile.

In that case, we have all learned median.

What was the median? Median gives us a central value.

Central value means that such a value or such a quartile where there 50% below data and 50% above data.

So, in the same way suppose if I have a median, which is my second quartile.

Whichever data will be 50% below in the distribution, I'll keep that in the left and the 50% above data that will come I will keep that in the right and we call this as Q2 which is the second quartile.

Now, what is Q3? We have seen 25% value which is my Q1.

We saw 50% value which is my Q2.

So, the 75% value that we see that is called by Q3.

75% means 75% of the data lies below the distribution and 25% of the data lies above the distribution.

So, in this particular scenario, when we have data divided into four equal parts, we'll call it as quartiles.

You can understand one more thing in it.

Let's see it through an example: what is a quartile? Suppose I have a data set, okay.

In which there are values from 10 to 100.

It is divided in the ratio of 10,10, 20,30, 40, in this way I have got one data set with me.

In this particular data set.

You tell me what is the range, the range would be maximum minus minimum value.

Which is my 100 minus 10.

So, my range in this particular scenario is 90.

Since I have even number of data points.

So, for me to first of all to find the median, I will have to find out the mean of the two central values, which means of the fifth and sixth value, which is 50 and 60.

Median comes as 55.

So, in this particular scenario, our Q2 is 55.

I have included 55 in my data set.

I have to calculate Q1 and Q3.

How will we find Q1? To find Q1 there is one more way which we can basically call that my left part data, in that particular part and my median, the actual median that I have calculated, which is Q2.

So, my median between these two, I will call that as Q1 or in simple words, Q1 is the median of the first half of the distribution.

So, we will want to see once how we will calculate Q1 In this scenario.

In this particular case, 10, 20, 30, 40, 50 and 55 is my first half of the distribution.

In this 30 would be my middle value.

And this will be my Q1.

The data on the right side of the median, that will help me in calculating the Q3.

We can ideally call Q3 as median of the second half of the distribution.

Like in this particular scenario 55, 60, 70, 80, 90, 100, these six values are there, in this whichever would be my median, that would be 80 because it is my central value.

So, in this particular case, 80 is my Q3.

And if we see in this total data set, my minimum value would be 10.

My Q1 would be 30, my median would be 55, Q3 is 80 and maximum value is 100.

So, in this way, we will calculate the quartiles.

So, till now we have covered about quartiles.

Now we will see one important topic which we call as percentile.

Before understanding percentile, we will see what are the percentages.

If we get marks in any exam then we calculate its percentage.

Suppose I have my five subject marks.

In the first subject, I got 89, in the second I got 92, in the third I got 87, in the fourth subject 95 and in the fifth subject 88.

So, we calculate a percentage of all these five subjects.

How do we calculate the percentage? Whatever marks that I've got, I have summed it up and divided it into a total number of subjects.

With this you have understood that you have got 90% marks in that particular term or in that exam.

With this I got to know that I have got 90% marks in one particular exam.

But with this I'm not able to get an idea how better I have performed compared to the other kids of my class or how bad I have performed.

So, to understand that concept, we will make use of percentiles.

So, basically what is percentile? It is one such value, below which how much data lies, that is told to us by percentile.

Suppose if I say that your score lies in 98 percentiles.

So, this means that you have performed better than 90% of kids in your class.

98 percentile means the 90% values which are there  below your value.

So, percentile we basically use for one comparison score.

So, you have your own score, which is 90 percentile and you have to compare how better you have performed from the rest of the group or the rest of the class so that we can compare through percentile.

Many universities and colleges use the percentile concept.

If you give competitive exams like GATE and CAT, then we use more of percentile.

Let's see with an example.

How do we calculate percentiles? Suppose I have a data set in which I have got the respective values in this way 2, 2, 3, 4, 5, 5, 5, 6, 7 and so on.

So, now I have been told to tell the percentile of the number 10.

What is the meaning of 10 percentile? It means that I have to tell that below 10 how many percent values come, which means 10 is above how many values.

To tell this we have got a very simple formula of percentile.

It goes this way, number of values below score, which means what is your score in this particular exam? It is 10.

So, we have to find how many values are below 10.

So, here are a number of values which are below my score.

I will be calculating that; I will divide it with the total number of scores and multiply it by 100.

So, if I put the values in this formula, the number of values that are below 10 are in total 16 values.

The total number of values that are there in my data set is 20 values.

If you sum them up, you will get it as 20.

Multiply it by 100.

I will get one percentile value which is 80.

This means that 10 lies above 80% in my distribution.

This means 80% more values lie below 10.

We can say this in many ways.

So, we just saw how to calculate percentiles of any value.

Suppose I've already given a percentile, then I have to find out on that percentile which value lies.

Which means we have to work on the reverse scenario.

What was the first case? It was to find the percentile of one value.

Now I have to find the percentile rank which means I have the percentile of that value in that data set.

And I have to find that value.

So, for that there is a formula for percentile rank which is in this way.

Percentile rank is equal to percentile upon 100 multiplied by n plus one.

Now what is this N? N would be my total number of items, in my entire data set the total numbers that are there, we replace n’s value with that.

We will see this with one example.

This is the old data set that we had already seen, in the same data set we have to find which value lies in the 25th percentile.

Which means I have to know which are the values below which there are 25% more values.

What will we do in this? We will put our values in the rank formula.

What is my percentile over here? It is 25 so 25 upon 100 multiplied by…how much was the total value with me? 20.

So my n’s value becomes 20, 20 +1= 21.

This we have put in the formula, when we calculated the value, I got one rank which is 5.25.

Now 5.25 doesn't have a basic significance.

Why? Because in my data set there is no value like 5.25.

So, this basically is called my index position.

The value that we get from the percentile formula.

That is called my index position.

Which means this is such a value, in front of it and behind it the values that are there, after finding that we can find our exact value.

To see that we will understand this further.

So, I have 5.25 as one index position.

It means that my value lies between five or six, between these two positions, okay.

So, I have got 2 values, one on the fifth position and one on the sixth position.

So, on this index position, this value that I have received from that if I want to find out the exact value.

So, we will find it with two steps.

What is the first step? Whatever is the decimal value in the index position, which means 0.25, we will take that value and we will multiply that, the difference that is there in both the fifth and sixth score.

Which means I took the decimal value from 5.25 which is 0.25 and the two values that I'll get, 5.25 lies between what? It lies either between five or six.

So, those two positions will be in my data set as 5 and 5.

So, the 5 minus 5 difference that is there.

That is 0.

So, in the first step I calculated whatever is my decimal value, I will multiply it with the difference whatever value has come I have noted it, the second step that is there, the lower score that I have, which is at the fifth position.

In that I added this particular value, which is 0 that I have calculated.

So, with that I will get my exact value, which means 25 percentile lies on which value?  So, we have just seen that our one percentile was already given.

So, how we have to find its value.

So, if you have values in any decimal.

So, we can follow two steps and find that value.

So, now we have learned both the concepts, quartiles as well as percentile but if you must have paid attention these two are very related to each other.

Basically, quartiles are special cases of percentiles.

Where quartile divides my data into four equal parts.

And percentile divides my ordered data into 100 equal parts.

To calculate both of these I will have to arrange my data into the smallest to the largest value or I have to sort it.

And my first, second and third quartiles that are there.

We also call it the 25th percentile, 50^th and 75^th percentile as well.

My first quartile will be called the 25th percentile, which means 25% of the values lie below the distribution and 75% lies above it.

the 50th percentile is basically my median which we also call the second quartile.

The third quarter means that 75% of my value in distribution lies below that particular.

Third and the important criteria which is in this particular lesson is inter quartile range.

Now we had seen that we had data, we calculated quartiles, we calculated mean.

But now you must be seeing one term “inter” which is in front of quartile and there is one more word “range”.

We have learned these three terms together.

We know the meaning of inter which is in between, quartile, we just saw it, range we already know it.

We will apply the understanding of all three into one concept and see what is inter quartile range.

What was the range? Range was giving me the spread of the whole data set.

What does inter quartile range give? It gives me the range between second and third quartile.

Basically, what is it? If from the maximum value, I subtract the minimum value then it is called my total range.

But if I subtract Q1 from Q3, that will be my inter quartile range.

So simply put, my middle half of the data which is from Q1 to Q3, if I calculate a range in between that then I will call it inter quartile range.

Okay.

Like in this particular scenario here 25%, 25, 25, 25, my data is divided into four quarters, Q1, Q2, Q3, these three are my values.

So, my IQR, which is InterQuartile Range, it will be my Q3 minus Q1.

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