Formula to Calculate Quartile in Statistics

The quartile formula is a statistical tool to calculate the variance from the given data by dividing the same into four defined intervals, comparing the results with the entire set of observations, and commenting on the differences in the data sets.

It is often used in statistics to measure the variances, which describe a division of all the given observations into four defined intervals based on the data values and where they stand when compared with the entire set of the given observations.

It divides into 3 points: a lower quartile, denoted by Q1, which falls between the smallest value and the median of the given data set. The median, denoted by Q2, is the median, and the upper quartile, denoted by Q3, is the middle point between the medianMedianThe median formula in statistics is used to determine the middle number in a data set that is arranged in ascending order. Median ={(n+1)/2}thread more and the highest number of the given distribution dataset.

Quartile Formula in statistics is represented as follows,

The Quartile Formula for Q1= ¼(n+1)th term The Quartile Formula for Q3= ¾(n+1)th term The Quartile Formula for Q2= Q3–Q1(Equivalent to Median)

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Explanation

The quartiles will divide the measurements of the given data set or sample into four similar or equal parts. For example, 25% of the given dataset’s measurements (represented by Q1) are not greater than the lower quartile. 50% of the measurements are not greater than the median, i.e., Q2; lastly, 75% will be less than the upper quartile that Q3 denotes. Therefore, 50% of the measurements of the given dataset are in between Q1, the lower quartile, and Q2, the upper quartile.

Examples

Let’s see some simple to advanced examples of a quartile in excelQuartile In ExcelQuartile functions are used to find the various quartiles of a data set and are part of Excel’s statistical functions. There are three quartiles; the first quartile (Q1) is the middle number between the smallest value and the median value of a data set. The second quartile (Q2) is the median of the data. The third quartile (Q3) is the middle value between the median of the data set and the highest value.read more to understand it better.

Example #1

Consider a data set of the following numbers: 10, 2, 4, 7, 8, 5, 11, 3, 12. You are required to calculate all the 3 quartiles.

Solution:

Use the following data for the calculation of quartile.

Calculation of Median or Q2 can be done as follows,

Median or Q2 = Sum(2+3+4+5+7+8+10+11+12)/9

Median or Q2 will be –

Median or Q2 = 7

Since the number of observations is odd, which is 9, the median would lie in the 5th position, which is 7, and the same will be Q2 for this example.

Calculation of Q1 can be done as follows,

Q1= ¼ (9 + 1)

= ¼ (10)

Q1 will be –

Q1 = 2.5

This means that Q1 is the average of 2nd and 3rd position of the observations, which is 3 & 4 here, and the average of the same is (3+4)/2 = 3.5

Calculation of Q3 can be done as follows,

Q3 = ¾ (9 + 1)

= ¾ (10)

Q3 will be –

Q3 = 7.5 Term

It means that Q3 is the average of the 8th and 9th positions of the observations, which is 10 and 11 here. So the same average is (10+11)/2 = 10.5.

Example #2

Simple Ltd. is a clothing manufacturer working on a scheme to please their employees for their efforts. The management is in discussion to start a new initiative which states they want to divide their employees as per the following:

  • Top 25% lying above Q3- $25 per clothGreater than the middle one but less than Q3 – $20 per clothGreater than Q1 but less than Q2 – $18 per clothThe management has collected its average daily production data for the last 10 days per (average) employee.55, 69, 88, 50, 77, 45, 40, 90, 75, 56.Use the quartile formula to build the reward structure.What rewards would an employee get if he produced 76 clothes ready?

Q1 = ¼ (n+1)th term

= ¼ (10+1)

= ¼ (11)

Q1 = 2.75 Term

Here, the average must be taken, which is of 2nd and 3rd terms, which are 45 and 50. The average formulaAverage FormulaAverage is the value that is used to represent the set of values of data as is the average calculated from whole data and this formula is calculated by adding all the values of the set given, denoted by summation of X and dividing it by the number of values given in set denoted by N.read more of the same is (45+50)/2 = 47.50

The Q1 is 47.50, which is bottom 25%

Calculation of quartile Q3 can be done as follows,

Q3 = ¾ (n+1)th term

= ¾ (11)

Q3 = 8.25 Term

Here, the average needs to be taken, which is of 8th and 9th terms, which are 88 and 90. The average of the same is (88+90)/2 = 89.00.

The Q3 is 89, which is the top 25%.

The calculation of the median or Q2 can be as follows:

The Median Value (Q2) = 8.25 – 2.75

Median or Q2= 5.5 Term

Here, the average needs to be taken, which is of 5th and 6th 56 and 69. The average of the same is (56+69)/2 = 62.5.

The Q2 or median is 62.5

Which is 50% of the population.

The Reward Range would be:

47.50 – 62.50 will get $18 per cloth

62.50 – 89 will get $20 per cloth

89.00 will get $25 per cloth

If an employee produces 76, he will lie above Q1. Hence, would be eligible for a $20 bonus.

Example #3

Teaching private coaching classes is considering rewarding students in the top 25% quartile advice to interquartile students lying in that range and retake sessions for the students lying below Q1.Use the quartile formula to determine what repercussions he will face if he scores an average of 63.

Solution :

The data is for the 25 students.

= ¼ (25+1)

= ¼ (26)

Q1 = 6.5 Term

The Q1 is 56.00, which is the bottom 25%

= ¾ (26)

Q3 = 19.50 Term

Here, the average needs to be taken, which is of 19th and 20th terms, which are 77 and 77. The average of the same is (77+77)/2 = 77.00.

The Q3 is 77, which is the top 25%.

Median or Q2=19.50 – 6.5

Median or Q2 = 13 Term

The Q2 or median is 68.00

The Range would be:

56.00 – 68.00

68.00 – 77.00

77.00

Relevance and Use of Quartile Formula

Quartiles let one quickly divide a given dataset or sample into four major groups, making it easy for the user to evaluate which of the four groups a data point is. While the median, which measures the central point of the dataset, is a robust estimator of the location. It does not say how much the data of the observations lie on either side or is dispersed or spread. The quartile measures the spread or dispersionDispersionIn statistics, dispersion (or spread) is a means of describing the extent of distribution of data around a central value or point. It aids in understanding data distribution.read more of values above and below the arithmetic meanArithmetic MeanArithmetic mean denotes the average of all the observations of a data series. It is the aggregate of all the values in a data set divided by the total count of the observations.read more or average, dividing the distribution into four major groups discussed above.

This article has been a guide to Quartile Formula. Here, we learn how to calculate quartiles in statistics using its formula, practical examples, and a downloadable Excel template. You can learn more about Excel modeling from the following articles: –

  • Point EstimatorsQuartile DeviationQuartile DeviationQuartile deviation is based on the difference between the first quartile and the third quartile in the frequency distribution and the difference is also known as the interquartile range, the difference divided by two is known as quartile deviation or semi interquartile range.read moreCoefficient of Variation FormulaCoefficient Of Variation FormulaThe coefficient of Variation is the systematized measure of a Probability Distribution’s or Frequency Distribution’s dispersion. It is determined as the ratio of Standard Deviation to the Mean. read moreCalculate the Correlation CoefficientCalculate The Correlation CoefficientCorrelation Coefficient, sometimes known as cross-correlation coefficient, is a statistical measure used to evaluate the strength of a relationship between 2 variables. Its values range from -1.0 (negative correlation) to +1.0 (positive correlation). read more