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1. GROUP - 3 ELEMENTS OF STATISTICS TOPIC : SUBMITTED BY : Aryan Pratap Singh R OLL NUMBER : 14 S UBMITTED TO : MOHD. FAHEEM SIR

2. Decile is a method that is used to divide a distribution into ten equal parts. When data is divided into deciles a decile rank is assigned to each data point in order to sort the data into point in order to sort the data into ascending or descending order. A decile has 10 categorical buckets while a quartile has 4 and a percentile has 100. The concept of a decile is used widely in the field of finance and economics to perform the analysis of data. It can be used to check the performance of a portfolio in the field of finance. In this article, we learn more about a decile , its definition, rank, and see associated examples on calculating the decile value.

3. What is Decile ? Decile , percentile, quartile, and quintile are different types of quantiles in statistics. A quantile refers to value that device the observations in a sample into equal subsection. There will always be 1 lesser quantile then the number of subsection created. A decile is a quantitative method of splitting up a set of ranked data into 10 equally large subsection. This type of data ranking is perform as part of many academic and statistical studies in the finance and economics fields. The data may be ranked from largest to smallest values, or vice versa. A decile , which has 10 categorical buckets may be contrasted with percentiles that have 100, quartiles that have four or quintiles that have five.

4. Decile Definition- Decile is a type of quantile that divides the data set into 10 equal subsections with the help of 9 data points. Each section of the sorted data represent of the original sample or population. Decile helps to order large amounts of data in the increasing or decreasing order. This ordering is done by using a scale from 1 to 10 where each successive value represents an increase by 10 percentage point.   Decile for Simple Data Decile is a method that is used to divide a distribution into ten equal parts. When data is divided into deciles a decile rank is assigned to each data point in order to sort the data into ascending or descending order. Formula:  

6. Example: We have the following data: 3,7,8,5,12,9,15,11,10,14,6,13,16,18,17,20,19,22,21,25 Sorted data: 3,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,25 Calculate the position for each decile : First Decile (D1):  

7. The 2nd value in the stored data is 5, so D1=5 Second Decile (D2): The 4th value in the stored data is 7, so D2=7   Inter Decile Range:  

8. The inter- decile range (IDR) is the difference between two deciles , which is D9 and D1. It represents the range of values that encompasses the middle 80% of the data.   IDR = D9 - D1   Where:   - D9 is the 9th decile   - D1 is the 1st decile     For example,  If D9 = 80 and D1 = 20, Then:  

9. This means that 80% of the data falls within the range of 20 to 80.   Semi Inter Decile Range or Decile Deviation:    The Semi Inter Decile Range (SIDR) is half of the Inter Decile Range (IDR). It's calculated as:       Where:   - D9 is the decile - D1 is the decile    

10. Example:   If D9 = 80 and D1 = 20, then:     Coefficient of Decile Deviation:   The Coefficient of Decile Deviation (CD) is a measure of relative variability that uses deciles (D1 and D9) to calculate the spread of a dataset. It's calculated as:    

11. Where:   - D9 is the 9th decile - D1 is the 1st decile   Example: Find the IDR, DD and CDD of the following frequency distribution: Class Interval  Frequency  0 -10 1 10 - 20 5 20 - 30 6 30 - 40 3 40 - 50 5

12. CI F CF 10 - 20 1 1 20 - 30 5 6 30 - 40 6 12 40 - 50 3 15 50 - 60 5 20 Solution: For D1:  N10 = 2010   = 2 So,  l1 = 10, l2 = 20, c = 1, f = 5

14. For D9:     So, l1 = 40, l2 = 50, c = 15, f = 5      

16.           Decile formula for grouped data : D(x)= I is the lower boundary of the class containing the decile given by , cf is the cumulative frequency of the entire data set, w is the size of the class, N is the total frequency, c is the cumulative frequency of the preceding class. The next section will cover the steps for calculation a particular decile .  

17. Decile Example - Suppose a data set consist of following numbers: 24, 32, 27, 32, 23, 62, 45, 80, 59, 63, 36, 54, 57, 36, 72, 55, 51, 32, 56, 33, 42, 55, 30. The value of the first two decide has to be calculated. The step required are as follows: Step 1: Arrange the data in increasing order. This gives 23, 24, 27, 30, 32, 32, 32, 33, 36, 36, 42, 45, 51, 54, 55, 55, 56, 57, 59, 62, 63, 72, 80. Step 2: Identify the total number of points. Here, n=23 Step 3: Apply the decile formula top calculate the position of the required data point. D (1) = . This implies the value of the 2.4 th data point has to be determined. This will lie between the score in the 2 nd and 3 rd position . In other words, the 2.4 th data is 0.4 of the way between the scores 24 and 27. Step 4: The value of the decile can be determined as [lower score+ (distance) (higher score-lower score)]. This is given as 24+0,4* (27-24) = 25.2  

18. Step 5: Apply steps 3 and 4 to determine the rest of the deciles . D (2) = =4.8 th data between digit number 4 and 5. Thus, 30+0.8* (32-30) = 31.6   Important Notes on Decile -   A decile is a quartile that is used to divide a data set into 10n equal subsections. The 5 th decile will be the median for the dataset. The decile formula for ungrouped data is given as term in the data set. The decile formula for grouped data is given by .  

19. Example 1: Find the 6 th and the 9 th decile for the data in the above-mentioned example. Solution: The arranged data is 23, 24, 27, 30, 32, 32, 32, 33, 36, 42, 45, 51, 54, 55, 55, 56, 57, 59, 62, 63, 72, and 80 N=23 D (6) = = 14.4 th data. This lies between 54 and 55. D (6) = = 54.4 D (9) = = 21.6 th data. This lies between 63 and 72 D (9) = 63+0.6* (72-63) = 68.4 Answer: D (5) = 54.4 and D (9) = 68.4  

20. 48 52 55 57 58 60 61 64 65 66   69 72 73 75 76 78 81 82 84   87 88 90 91 92 93 94 95 96 97 99   Example of a Decile - The table below show the ungrouped scores (out of 100) for 30 exam takers:

21. Using the information presented in the table, the 1 st decile can be calculated as: Value of data Value of 3.1 st data, which is 0.1 of the way between scores 55 and 57 55+2(0.1) = 55.2 = D1 D1 means that 10% of data set falls below 55.2 Let’s calculate the third decile : D3= value of term D3 = value of 9.3 rd position, which is 0.3 between the scores of 65 and 66 Thus, D3 = 65+1(0.3)= 65.3 3% of the score 30 scores in the observation fall below 65.3  

22. What would be get if we were to calculate the 5 th decile ? D5= value of D5= value of 15.5 th position, half way between scores 76 and 78 50% of the scores fall below 77 Also, notice how the 5 th decile is also the median of the observation. Looking at the data set in the table, the median, which is the middle data point of any given set of numbers, can be calculated as = 77 = median = D5. At this point, half of the scores lie above and below the distribution.  

23. Decile for continuous data : In statistics, a decile quartile that divides a data set into 10 equal parts, with each part containing the same amount of data. Deciles are often used to summarize a data set and identify values that separate the data into different percentiles. Deciles can only be calculated when the data is in ascending or descending order. The value of the decile can be affected by outliers and extreme values in the data set Formula of decile for continuous data: D1 =L1+   D2 =L2+    

24. D3 =L3+   D4 =L4+   D5 =L5+   D6 =L6+   D7 =L7+ D8 =L8+  

25. D9 =L9+   For Example:   Example 1- Find the 6 th and 8 th decile for the data in the above – mentioned example. Solution: The arranged data is 23 , 24 , 27 , 30 , 32 , 32 , 32 , 33 , 36 , 36 , 42 , 45 , 51 , 54 , 55 , 55 , 56 , 57 , 59 , 62 , 63 , 72 , 80  

26. n=23 D6 = 14.4 th data. This lies between 54 and 55 D6 = 54 + 0.4* (55-54) = 54.4 D9 = = 21.6 th data . This lies between 63 and 72 D9 = 63 +0.6*(72-63) = 68.4 Answer: D5=54.4 and D9=68.4 2. The table below shows the ungrouped scores (out of 100) for 30 exam takers: 40 52 55 57 58 60 61 64 65 66  

27. 69 72 73 75 76 78 81 82 84 87 88 90 91 92 93 94 95 96 97 99 Using the information presented in the table, the 1 st decile can be calculated as: . =value of data   . = value of 3.1 st data, which is 0.1 of the way between scores 55 and 57 . = 55+2(0.1) = 55.2 = D1 . = D1 means that 10% of the data set falls below 55.2. Let’s calculate the 3 rd decile . D3 = value of . D3 = value of 9.3 rd position, which is 0.3 between the scores of 65 and 66 . Thus, D3 = 65 + 1 (0.3) = 65.3 What would we get if we were to calculate the 5 th decile ? . D5 = value of  

28. . D5 = value of 15.5 th position, halfway between scores 76 and 78 . D5 = 50% of the scores fall below 77 Also notice how the 5 th decile is also the median of the observation. Looking at the dataset in the table, the median which is the middle data point of any given set of numbers, can be calculated as .At this point, half of the scores lie above and below the distribution.   Percentile: In statistics, a ‘percentile’ is a measure that indicates the value below which a given percentage of observation in a data set fall. For example: The 25 th percentile (also known as first quartile) is the value below which 25% of the observation may be found. How’s percentile works:

29. th percentile: the minimum value in the data set. 50 th percentile (median): the middle value, where 50% observation are beloe and 50% are above. 100 th percentile: the maximum value in the data set. Percentiles are often used to understand the relative standing of a particular value within a data set. Ex: if a test score is at the 90 th percentile, this means the score is higher than 90% of the score of those who took the test. Features of percentile: Here’s some key features of percentile are as follows: