Statistics for Data Analyst

Quartile deviation denoted by QD is the absolute measure of dispersion and it is defined as the half of the difference between the upper quartile ($Q_3$) and the lower quartile ($Q_1$).

The Quartile Deviation also known as semi-interquartile range (Semi IQR), is a measure of dispersion that focuses on the middle 50% of the data. It is calculated as half the difference between the Third Quartile ($Q_3$) and the First Quartile ($Q_1$). One can write it mathematically as

$$QD = \frac{Q_3-Q_1}{2}$$

Note that the interquartile range is only the difference between the upper quartile ($Q_3$) and the lower quartile ($Q_1$), that is,

$$Interquartile\,\, Range = IRQ = Q_3 – Q_1$$

The Relative Measure of Quartile Deviation is the Coefficient of Quartile Deviation and is given as

$$Coefficient\,\,of\,\,QD = \frac{Q_3 – Q_1}{Q_3 + Q_1}\times 100$$

When to Use QD

• When dealing with skewed data or data with outliers.
• When a quick and easy measure of dispersion is needed.

Interpretation QD

Spread: A larger quartile deviation indicates greater variability in the middle portion of the data.
Outliers: QD is less sensitive to extreme values (outliers) compared to the standard deviation.

Quartile Deviation for Ungrouped Data

22	22	25	25	30	30	30	31	31	33	36	39
40	40	42	42	48	48	50	51	52	55	57	59
81	86	89	89	90	91	91	91	92	93	93	93
93	94	94	94	95	96	96	96	97	97	98	98
99	99	99	100	100	100	101	101	102	102	102	102
102	103	103	104	104	104	105	106	106	106	107	108
108	108	109	109	109	110	111	112	112	113	113	113
113	114	115	116	116	117	117	117	118	118	119	121

The above data is already sorted and there are a total of 96 observations. The first and third quartiles of the data can be computed as follows:

$Q_1 = \left(\frac{n}{4}\right)th$ value $= \left(\frac{96}{4}\right)th$ value $= 24th$ value. The 24th observation is 59, therefore, $Q_1=59$.

$Q_3 = \left(\frac{3n}{4}\right)th$ value $= \left(\frac{3\times 96}{4}\right)th$ value $= 72th$ value. The 72nd observation is 108, therefore, $Q_3=108$.

The quartile deviation will be

$$QD=\frac{Q_3 – Q_1}{2} = \frac{108-59}{2} = 24.5$$

The Interquartile Range $= IQR = Q_3 – Q_1 = 108 – 59 = 49$

The coefficient of Quantile Deviation will be

$$Coefficient\,\, of\,\, QD = \frac{Q_3 – Q_1}{Q_3 – Q_1}\times 100 = \frac{108-59}{108+59}\times 100 = 29.34\%$$

Quartile Deviation for Grouped Data

Consider the following example for grouped data to compute the quartile deviation.

Classes	Frequencies	Class Boundaries	CF
11-14.9	11	10.95-14.95	11
15-20.9	19	14.95-20.95	30
21-24.9	21	20.95-24.95	51
25-30.9	34	24.95-30.95	85
31-34.9	16	30.95-34.95	101
35-40.9	9	34.95-40.95	110
41-44.9	4	40.95-44.95	114
Total	114

The first and third quartiles for the above-grouped data will be

\begin{align*}
Q_1 &= l + \frac{h}{f}\left(\frac{n}{4} – C\right)\\
&= 14.95 + \frac{6}{19}\left(\frac{114}{4} – 11\right)\\
&= 14.95 + \frac{6}{19}(28.5 – 11) = 20.48\\
Q_3 &= l + \frac{h}{f}\left(\frac{3\times 114}{4}-85\right)\\
&=30.95 + 0.187418 = 31.14
\end{align*}

The QD is

$$QD = \frac{Q_3 – Q_1}{2} = \frac{31.14 – 20.48}{2} = \frac{10.66}{2} = 5.33$$

The Interquartile Range will be

$$IQR = Q_3 – Q_1 = 31.14 – 20.48 = 10.66$$

The coefficient of quartile deviation is

$$Coefficient\,\,of\,\, QD = \frac{Q_3 – Q_1}{Q_3 + Q_1}\times 100 = \frac{31.14 – 20.48}{31.14+20.48}\times 100 = 20.65\%$$

Advantages of QD

• Less affected by outliers: Makes it suitable for skewed data.
• Easy to calculate: Relatively simple compared to standard deviation.

Disadvantages of QD

• Ignores extreme values: This may not provide a complete picture of the data’s spread.
• Less sensitive to changes in data: Compared to standard deviation.

In summary, Quartile deviation is a valuable and useful tool for understanding the spread of data, particularly when outliers are present. By focusing on the middle 50% of the data, it provides a robust measure of dispersion that is less sensitive to extreme values. However, it is important to consider its limitations, such as its insensitivity to outliers and changes in data.

Frequently Asked Questions about Quartile Deviation

1. What is quartile deviation?
2. What are the advantages of QD?
3. What are the disadvantages of QD?
4. What is IQR?
5. What is Semi-IQR?
6. How QD is interpreted?
7. How QD is computed for grouped and ungrouped data?
8. When QD should be used?

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