quartile deviation: An introduction

One of the three points that divide a data set into four equal parts. Or the values that divide data into quarters. Each group contains equal number of observations or data. Median acts as base for calculation of quartile.

1. Quartile Deviation Dr Rajesh Verma Asst. Prof ( Psychology ) Govt. College Adampur, Hisar, Haryana

2. Quartile: Meaning One of the three points that divide a data set into four equal parts. Or the values that divide data into quarters. Each group contains equal number of observations or data. Median acts as base for calculation of quartile.

3. Quartile Deviation: Definition The quartile deviation is half of the difference between first quartile (Q1) and third quartile (Q3). This is also known as quartile coefficient of dispersion. QD = “A measure of dispersion that is defined as the value halfway between the first and third quartiles (i.e., half the interquartile range). Also called semi-interquartile range” (APA). Garret (2014) defines, “the Quartile deviation or Q is half the scale distance between 75th and 25th percent is a frequency distribution”. According to Guilford (1963) the Semi inter Quartile range Q is the one half the range of the middle 50 percent of the cases.  

4. Explanation So, this way we have three quartiles i.e. Q1, Q2 and Q3. (i) Q1 – It is the midpoint of lowest 50% of data and also known as Lowest quartile or first quartile. (ii) Q2 – It is the median of the data or the middle point of a given data set and also known as second quartile. (iii) Q3 – It is the midpoint of highest 50% of data and also known as highest quartile or third quartile. Thus the quartile measures the dispersion of score above and below the median by dividing the entire data set into four equal groups.

5. Introduction Any data set that meets the assumptions of normal distribution tends to have maximum number of values in the middle. It’s a kind of half of the range of middle 50% data and also known as Semi inter-quartile range. Quartile deviation (Q) is absolute measure of dispersion of the central or middle portion of the data set. Important Note: The students must understand the difference between Quartile and Quarter. Quartile is a point on a data set where as Quarter is 1/4 th part. You can be in a quarter but you have to be at a quartile .

6. Characteristics of QD 1. Median is base of the quartile deviation. 2. Quartile deviation is not affected with extreme values. 3. In a symmetrical distributions Q1 and Q3 are at equal distance from the median (Median-Q1=Q3-Median). 4. QD is the best measure of variability for open ended distribution. 5. Quartiles are three points on the distribution that divides the distribution into 4 quarters.

7. 6. The Q1 and Q3 are lower and upper limits of middle 50% distribution. 7. It is the index of score density at the middle of distribution. 8. The larger the variability in a distribution the larger the value of Q and vice-versa. 9. In normal distribution Quartile distribution (Q) is called probability error (PE).

8. Computation of QD For Ungrouped Data(Hypothetical data) ( i ) If data is in odd number Ex – 12, 54, 32, 51, 24, 60, 21, 44, 31, 48, 50 Step I – Arrange the raw data in ascending order. Therefore, 12, 21, 24, 31, 32, 44, 48, 50, 51, 54, 60 Step II – Find out Q1 Q1 = th position in the ordered distribution. therefore, Q1=11+1/4 = 3 rd position i.e. 24  

9. (12, 21, 24, 31, 32, 44, 48, 51, 54, 60) Step III – Find out Q3 Q3 = th position in the ordered distribution. therefore, Q3 = (11+1)3/4 = 9 th position i.e. 51 Step IV – Find out Semi-quartile range or QD Q = therefore, = = 27/2=   13.5

10. Computation of QD For Ungrouped Data(Hypothetical data) ( i ) If data is in even number Ex – 12, 54, 32, 51, 24, 60, 21, 44, 31, 48 Step I – Arrange the raw data in ascending order. Therefore, 12, 21, 24, 31, 32, 44, 48, 51, 54, 60 Step II – Find out Q1 Q1 = th position in the ordered distribution. therefore, Q1=11/4 = 2.75 th position i.e. 2 nd obs + .75 (3 rd obs -2 nd obs ), 21+.75(24-21) = 21+ 1.5 = 22.5  

11. (12, 21, 24, 31, 32, 44, 48, 50, 51, 54, 60) Step III – Find out Q3 Q3 = th position in the ordered distribution. therefore, Q3 = (10+1)3/4 = 8.25 th position i.e. 8 th obs + .25(9 th obs – 8 th obs ) = 51+.25(54-51) = 51+.25(3) => 51+.75 = 51.75 Step IV – Find out Semi-quartile range or QD Q = therefore, = 29.25/2 = 14.625  

12. For Grouped Data (Hypothetical Data) Step I – Find = 50/4 = 12.5, therefore, Class interval in which quartile Q1 falls is 20-24 Step II – Find Q1 Where, l = the exact lower limit of the interval in which the quartile falls. i = size of the class interval = Cumulative frequency of the previous class interval to the class interval in which quartile falls. fq = the frequency of the class interval containing the quartile. n = total number of observations or sum of frequencies.  

13. Calculation: - l = 19.5, i = 5, = 10, fq = 6, n = 50 Substituting the values, = = = = = = 5.33  

14. Step III – Find = = 150/4 = 37.5 therefore, Class interval in which quartile Q3 falls is 35-39 Step IV – Find Q3 with the following formula  

15. Calculation: - l = 34.5, i = 5, = 35, fq = 6, n = 50 Q Substituting the values, = = = = = = 7.83  

16. Substituting the values in the formula, = 1.25 So, our quartile deviation of our hypothetical grouped data is 1.25.  

17. References: 1. https://dictionary.apa.org/quartile-deviation. 2. Guilford, J. P. and Fruchter , B. (1978). Fundamental Statistics in Psychology and Education, 6th ed. Tokyo: McGraw-Hill. 3. https://todayinsci.com/M/Mahalanobis_Prasanta/ MahalanobisPrasanta-Quotations.htm. 4. Garrett, H. E. (2014). Statistics in Psychology and Education. New Delhi: Pragon International. 5. Levin, J. & Fox, J. A. (2006). Elementary Statistics. New Delhi: Pearson.

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