Q 606. Explain Top Quartile, Middle Quartile and Bottom Quartile quality scores and their usage with suitable examples.

 

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Quartile is the term used in statistics for dividing the data of observations into 4 defined intervals each having 25% of the data points by organizing data into 3 points as shown below:

 

Q1 - Mid data point between LD and median

Q2 - Mid data point between LD & HD

Q3 - Mid data point between HD and median

 

 

For eg- Data set considered for 15 days. Position of quartiles are identified in the data set of observations (ascending order) as shown below.

 

 

 

Quartiles and intervals for the sample of data observations of payments done are shown below:

 

 

Quartiles are three values that split your dataset into quarters. The three quartiles are the top quartile (Q3), the middle quartile (Q2, also known as the median), and the bottom quartile (Q1). They are used to understand the distribution of data, identify outliers, and summarize the central tendency and spread of a dataset. Quartiles are commonly used in data analysis, visualization, and identifying data points that fall outside the typical range.

 

 

The bottom quartile, Q1, is the 25th percentile of a dataset. It represents the data point below which 25% of the data falls. Example a dataset of QA scores for 100 agents. If Q1 is 60, it means that 25% of the agents scored below 60 QA score in quality/call monitoring.

The middle quartile, Q2, is the 50th percentile of a dataset, also known as the median. It represents the midpoint of the data, where half of the values are above, and half are below. Example in same dataset of QA Scores of 100 agents if Q2 (median) is 75, it means that 50% of the agents scored below 75 QA score, and 50% scored above 75 QA score.

The top quartile, Q3, is the 75th percentile of a dataset. It represents the data point below which 75% of the data falls. Example In the QA score dataset, if Q3 is 90, it means that 75% of the agents scored below 90 in call monitoring.

Also, in typical call center scenario audit sampling targets are determined based on qualities. For example, the agents falling in bottom quartile for their QA scores will have higher audit samples (3-4 audits per agent/ per week) as compared to the mid & bottom performers (1-2 audits per agent/ per week)

 

• Quartile measures the location of data in Descriptive Statistics and a modified form of Percentile.
• To calculate it we must first arrange data in increasing order.
• We must first understand Median if we want to comprehend Quartile better. Median can be perceived as the middle value of a data set or a number that segregates the ordered data into 2 equal halves. This means that half the values are the same or smaller than the median and the other half are either equal or larger.   
• Let us look at the following set of observations:                                                1 ; 11.5 ; 6 ; 7.2 ; 4 ; 8 ; 9 ; 10 ; 6.8 ; 8.3 ; 2 ; 2 ; 10 ; 1
• Let us 1st arrange it in ascending order:                                                                                 1 ; 1 ; 2 ; 2 ; 4 ; 6 ; 6.8 ; 7.2 ; 8 ; 8.3 ; 9 ; 10 ; 10 ; 11.5
• Since the data has 14 observations, the median will be between the 7^th and the 8^th values, that is, between 6.8 and 7.2 . The formula for median calculation = (6.8+7.2)/2 = 7. This means that half of the observations are equal to or larger than 7 while the other half equal or smaller.
• Quartiles segregate the data into quarters. The 1^st Quartile, Q₁, is the middle observation of the lower half of the data and the 3^rd Quartile is the middle observation of the upper half of the data.
• Let’s consider the same data set to explain the same :                                                       1 ; 1 ; 2 ; 2 ; 4 ; 6 ; 6.8 ; 7.2 ; 8 ; 8.3 ; 9 ; 10 ; 10 ; 11.5
• As the median or the 2^nd Quartile is 7, the lower half of the data are :                            1 ; 1 ; 2 ; 2 ; 4 ; 6 ; 6.8
• The middle value of the lower half of 2 and hence, it is the 1^st Quartile which means that 1/4^th of the entire set of observations is either equal to or less than 2 and that 3/4^th of the set of observations is larger than 2.
• The Quartile score of the value 2 is also 25% which means that 25% of the entire set of observations is equal to less than 2. In other words, Quartile score is a percentile rank and it is used to compare the performance of a data point to the rest of the data set.
• The upper half of the set of operations is :                                                                     7.2 ; 8 ; 8.3 ; 9 ; 10 ; 10 ; 11.5
• The 3^rd Quartile, Q₃, is 9. This means that the Quartile Score of 9 is 75%. In other words, 75% of the entire set of observations is less than 9.

 

 

 

Quartiles are used to categorize data points based on their position on a data set. Data is divided into 4 intervals based on values of data and how they compare to other data points. Each interval represents 25% of data points. It is analogous to median with the difference being median divides the data into 2 halves (50% each) while quartiles divides the data into 4 intervals (25% each). 

 

1. Top quartile 
    Also called as Upper quartile
    Contains top 25% of data
    It is 75th percentile of data

 

2. Middle quartile
    Also called as second quartile
    It is the 50th percentile of data
    It shows the central tendency of data

 

3. Bottom quartile
    Also called as lower quartile
    Contains bottom 25% of data

 

Usage of quartiles:

1. Helps understand distribution of data without being influenced by extreme values (outliers)
2. Helpful in finding outliers
3. Helps to understand central tendency and distribution

 

Quartiles divide the data into 4 intervals:

 

First interval - Contains data points between minimum value and bottom quartile
Second interval - Contains data points between bottom quartile and median
Third interval - Contains data points between median and top quartile
Fourth interval - Contains data points between top quartile and maximum value

 

Box plot best represents the quartiles. A sample box plot is given below

 

 

How to locate position of quartiles:

 

First operation to be done before using the formula is to rank / sort the data.

 

For odd number of data points:

 

Bottom quartile (Q1) = (N+1) * 1/4 
Middle quartile (Q2) = (N+1) * 2/4 
Upper quartile (Q3) = (N+1) * 3/4 

where N is count of data points. 

 

For even number of data points:

 

Bottom quartile (Q1) = median in lower half of data (not including median)
Middle quartile (Q2) = average of middle two values 
Upper quartile (Q3) = median in upper half of data (not including median)

 

Outliers:

 

Interquartile range = Q3 - Q1
Outlier = 1.5 * IQR (any data higher / lower than this value)

 

Example:

 

Person ABCDE wants to invest in stock market. He looks at the annual return % as the only evaluation parameter - data is shown below:

 

 

First step is to sort the data

 

 

Box plot is as follows

 

 

Bottom quartile = (7+1)* 1/4 = 2nd data point from bottom (-20% returns) 

Middle quartile = (7+1)* 2/4 = 4th data point (0% returns)

Top quartile = (7+1)* 3/4 = 2nd data point from top (50% returns)

 

Returns of company D is above top quartile. Hence person ABCDE decides to invest in company D. Also, company E is to be avoided since the returns are below bottom quartile.

Before understanding the quartile, let us understand the percentile concept which provides detailed information about how data are spread over the internal from the smallest value to the largest value. it is calculated by Px=x(n+1) /100 ; where Px is the value at which x percentage of data lie below the value and n is the total number of observation.

Suppose we have 21 solar modules power data and want to check the 25th, 50th and 75th percentile value from data then here is the answer by using formula:

 

25th percentile is nothing but Bottom quartile which means 25% data points are below in your rank ordered sequence.

75th percentile indicates to Top quartile  which means 75% data points below rank ordered sequence.

50th percentile belongs to Middle quartile which means half of the data lies above and half below.

 

Below graphical summary of the same data represents quartile score as stated above. it is used to determine the data central tendency, skewness etc.

 

 

While all respondents have accurately described the concept of quartiles and their calculation. However the question was oriented specially towards the quality scores and the use of concept of quartiles in it (might want to refer to Bottom quartile management). In this aspect, Arvind Swarup has provided the best answer and hence is the winner for this question.

 

P.S. Do read all answers to get a perspective on how quartiles are calculated.