Measure of Dispersion

The Empirical Rule (also known as the 68-95-99.7 Rule) is a statistical principle that applies to normally distributed data (bell-shaped curves). Empirical Rule tells us how data is spread around the mean in such (bell-shaped) distributions.

Empirical Rule states that:

• 68% of data falls within 1 standard deviation ($\sigma$) of the mean ($\mu$). In other words, 68% of the data falls within ±1 standard deviation ($\sigma$) of the mean ($\mu$). Range: $\mu-1\sigma$ to $\mu+1\sigma$.
• 95% of data falls within 2 standard deviations ($\sigma$) of the mean ($\mu$). In other words, 95% of the data falls within ±2 standard deviations ($2\sigma$) of the mean ($\mu$). Range: $\mu-2\sigma$ to $\mu+2\sigma$.
• 99.7% of data falls within 3 standard deviations ($\sigma$) of the mean ($\mu$). In other words, 99.7% of the data falls within ±3 standard deviations ($3\sigma$) of the mean ($\mu$). Range: $\mu-3\sigma$ to $\mu+3\sigma$.

Visual Representation of Empirical Rule

The empirical rule can be visualized from the following graphical representation:

Key Points

• Empirical Rule only applies to normal (symmetric, bell-shaped) distributions.
• It helps estimate probabilities and identify outliers.
• About 0.3% of data lies beyond ±3σ (considered rare events).

Numerical Example of Empirical Rule

Suppose adult human heights are normally distributed with Mean ($\mu$) = 70 inches and standard deviation ($\sigma$) = 3 inches. Then:

• 68% of heights are between 67–73 inches ($\mu \pm \sigma \Rightarrow 70 \pm 3$ ).
• 95% are between 64–76 inches ($\mu \pm 2\sigma\Rightarrow 70 \pm 2\times 3$).
• 99.7% are between 61–79 inches ($\mu \pm 3\sigma \Rightarrow 70 ± 3\times 3$).

This rule is a quick way to understand variability in normally distributed data without complex calculations. For non-normal distributions, other methods (like Chebyshev’s inequality) may be used.

Real-Life Applications & Examples

• Quality Control in Manufacturing: Manufacturers measure product dimensions (e.g., bottle fill volume, screw lengths). If the process is normally distributed, the Empirical Rule helps detect defects: If soda bottles have a mean volume of 500ml with $\sigma$ = 10ml:
  • 68% of bottles will be between 490ml–510ml.
  • 95% will be between 480ml–520ml.
  • Bottles outside 470ml–530ml (3$\sigma$) are rare and may indicate a production issue.

• Human Height Distribution: The Heights of people in a population often follow a normal distribution. If the average male height is 70 inches (5’10”) with $\sigma$ = 3 inches:
  • 68% of men are between 67–73 inches.
  • 95% are between 64–76 inches.
  • 99.7% are between 61–79 inches.

• Test Scores (Standardized Exams): The exam scores (SAT, IQ tests) are often normally distributed. If SAT scores have $\mu$ = 1000 and $\sigma$ = 200:
  • 68% of students score between 800–1200.
  • 95% score between 600–1400.
  • Extremely low (<400) or high (>1600) scores are rare.

• Financial Market Analysis (Stock Returns): The daily stock returns often follow a normal distribution. If a stock has an average daily return of 0.1% with σ = 2%: If a stock has an average daily return of 0.1% with σ = 2%:
  • 68% of days will see returns between -1.9% to +2.1%.
  • 95% will be between -3.9% to +4.1%.
  • Extreme crashes or surges beyond ±6% are very rare (0.3%).

• Medical Data (Blood Pressure, Cholesterol Levels): Many health metrics are normally distributed. If the average systolic blood pressure is 120 mmHg with $\sigma$ = 10:
  • 68% of people have readings between 110–130 mmHg.
  • 95% fall within 100–140 mmHg.
  • Readings above 150 mmHg may indicate hypertension.

• Weather Data (Temperature Variations): The daily temperatures in a region often follow a normal distribution. If the average July temperature is 85°F with σ = 5°F:
  • 68% of days will be between 80°F–90°F.
  • 95% will be between 75°F–95°F.
  • Extremely hot (>100°F) or cold (<70°F) days are rare.

Why the Empirical Rule Matters

• It helps in predicting probabilities without complex calculations.
• It is used in risk assessment (finance, insurance).
• It guides quality control and process improvements.
• It assists in setting thresholds (e.g., medical diagnostics, passing scores).

FAQs about Empirical Rule

• What is the empirical rule?
• For what kind of probability distribution, the empirical rule is used.
• What is the area under the curve (or percentage) if data falls within 1, 2, and 3 standard deviations?
• Represent the rule graphically.
• Give real-life applications and examples of the rule.
• Why the empirical rule matters, describe.

R Frequently Asked Questions

The importance of dispersion in statistics cannot be ignored. The term dispersion (or spread, or variability) is used to express the variability in the data set. The measure of dispersion is very important in statistics as it gives an average measure of how much data points differ from the average or another measure. The measure of variability tells about the consistency in the data sets.

The dispersion is a quantity that is far away from its center point (such as average). The data with minimum variation/variability with respect to its center point (average) is said to be more consistent. The lesser the variability in the data the more consistent the data.

Example of Measure of Dispersion

Suppose the score of three batsmen in three cricket matches:

Player	Match 1	Match 2	Match 3	Average Score
A	70	80	90	80
B	75	80	95	80
C	65	80	95	80

The question is which player is more consistent with his performance.

In the above data set the player whose deviation from average is minimum will be the most consistent player. So, the player B is more consistent than others. He shows less variation.

There are two types of measures of dispersion:

Absolute Measure of Dispersion

In absolute measure of dispersion, the measure is expressed in the original units in which the data is collected. For example, if data is collected in grams, the measure of dispersion will also be expressed in grams. The absolute measure of dispersion has the following types:

• Range
• Quartile Deviation
• Average Deviation
• Standard Deviation
• Variance

Relative Measures of Dispersion

In the relative measures of dispersion, the measure is expressed in terms of coefficients, percentages, ratios, etc. It has the following types:

• Coefficient of range
• Coefficient of Quartile Deviation
• Coefficient of Average Deviation
• Coefficient of Variation (CV)

See more about Measures of Dispersion

Range and Coefficient of Range

Range is defined as the difference between the maximum value and minimum value of the data, statistically, it is $R=x_{max} – x_{min}$.

The Coefficient of Range is $=\frac{x_{max} – x_{min} }{x_{max} – x_{min} }$. Multiplying it by 100 will express it in percentages.

Consider the ungrouped data $x = 32, 36, 36, 37, 39, 41, 45, 46, 48$

The range will be $x_{max} – x_{min} = 48 – 32 = 16$.

The coefficient of Range will be $=\frac{x_{max} – x_{min} }{x_{max} – x_{min} }$

\begin{align*}
Coef\,\, of\,\, Range =\frac{x_{max} – x_{min} }{x_{max} – x_{min} } \\
&= \frac{48-32}{48+32} = \frac{16}{80} = 0.2\\
&= 0.2 \times 100 = 20\%
\end{align*}

For the following grouped data, the range and coefficient of the range will be

Classes	Freq	Class Boundaries
65 – 84	9	64.5 – 84.5
85 – 104	10	84.5 – 104.5
105 – 124	17	104.5 – 124.5
125 – 144	10	124.5 – 144.5
145 – 164	5	144.5 – 164.5
165 – 184	4	164.5 – 184.5
185 – 204	5	184.5 – 204.5
Tota.	60

The upper class bound of the highest class will be $x_{min}$ and the lower class boundary of the lowest class will be $x_{min}$. Therefore, $x_{max}=204.5$ and $x_{min} = 64.5$. Therefore,

$$Range = x_{max} – x_{min} = 204.5 – 64.5 = 140$$

The Coefficient of Range will be

\begin{align*}
Coef\,\, of\,\, Range &=\frac{x_{max} – x_{min} }{x_{max} – x_{min} } \\
&= \frac{204.5-64.5}{204.5+64.5} = \frac{140}{269} = 0.5204\\
&= 0.5204 \times 100 = 52.04\%
\end{align*}

Average Deviation and Coefficient of Average Deviation

The average deviation is an absolute measure of dispersion. The mean/average of absolute deviation either taken from mean, median, or mode is called average deviation. Statistically, it is

$$Mean\,\, Deviation_{\overline{X}} = \frac{\sum\limits_{i=1}^n|x_i-\overline{x}|}{n}$$

$X$	$x-\overline{x}$	$|x-\overline{x}|$	$x-\tilde{x}$	$|x-\tilde{x}|$	$x-\hat{x}$	$|x-\hat{x}|$
32	$32-40 = -8$	8	$32-39=-7$	7	$32-36=-4$	4
36	$36-40=-4$	4	$36-39=-3$	3	$36-36=0$	0
36	$36-40=-4$	4	$36-39=-3$	3	$36-36=0$	0
37	$37-40=-3$	3	$37-39=-2$	2	$37-36=1$	1
39	$39-40=-1$	1	$39-39=0$	0	$39-36=3$	3
41	$41-40=1$	1	$41-39=2$	2	$41-36=5$	5
45	$45-40=5$	5	$45-39=6$	6	$45-36=9$	9
46	$46-40=6$	6	$46-39=7$	7	$46-36=10$	10
48	$48-40=8$	7	$48-39=9$	9	$48-36=12$	12
Total	0	40		39		36

Where
\begin{align*}
Mean &= \overline{x} = \frac{\sum\limits_{i=1}^n x_i}{n} = \frac{360}{9} = 40\\
Mode &= 36\\
Median &= 39\\
MD_{\overline{x}} &= \frac{\sum\limits_{i=1}^n |x-\overline{x}|}{n} = \frac{40}{9} = 4.44\\
MD_{\tilde{x}} &= \frac{\sum\limits_{i=1}^n |x-\tilde{x}|}{n} = \frac{39}{9} = 4.33\\
MD_{\hat{x}} &= \frac{\sum\limits_{i=1}^n |x-\hat{x}|}{n} = \frac{36}{9} = 4.00
\end{align*}

The relative measure of average deviation is the coefficient of average deviation. It can be calculated as follows:

Coefficient of Average Deviation from Mean (also called Mean Coefficient of Dispersion)

\begin{align*}\text{Mean Coefficient of Dispersion} = \frac{MD_{\overline{x}}}{\overline{x}} = \frac{4.44}{40}\times 100 = 11.1\%\end{align*}

Coefficient of Average Deviation from Median (also called Median Coefficient of Dispersion)

\begin{align*}\text{Median Coefficient of Dispersion} = \frac{MD_{\tilde{x}}}{\tilde{x}} = \frac{4.33}{39}\times 100 = 11.1\%\end{align*}

Coefficient of Average Deviation from Mode (also called Mode Coefficient of Dispersion)

\begin{align*}\text{Mode Coefficient of Dispersion} = \frac{MD_{\hat{x}}}{\hat{x}} = \frac{4}{36}\times 100 = 11.1\%\end{align*}

Average Deviation for Grouped Data

One can also compute average deviations for grouped data (Discrete Case) as follows:

$x$ Mid Point	$f$	$fx$	$|x-\overline{x}|$	$f|x-\overline{x}|$	$|x-\tilde{x}|$	$f|x-\tilde{x}|$
10	9	90	$10-34=24$	216	20	180
20	10	200	$20-34=14$	140	10	100
30	17	510	$30-34=4$	68	0	0
40	10	400	$40-34=6$	60	10	100
50	5	250	$50-34=16$	80	20	100
60	4	240	$60-34=26$	104	30	120
70	5	350	$70-34=36$	180	40	200
Total	60	2040		848		800

\begin{align*}
\overline{x} &= \frac{\sum\limits_{i=1}^n}{n} = \frac{2040}{60} = 34\\
\tilde{x} &= 30\\
\hat{x} &= 30\\
MD_{\overline{x}} &= \frac{\sum\limits_{i=1}^n f|x-\overline{x}|}{n} = \frac{848}{60} = 14.13\\
MD_{\tilde{x}} &= \frac{\sum\limits_{i=1}^n f|x-\tilde{x}|}{n} = \frac{800}{60} = 13.33\\
MD_{\hat{x}} &= \frac{\sum\limits_{i=1}^n |x-\hat{x}|}{n} = \frac{36}{9} = 4\\
\text{Mean Coefficient of Dispersion} &= \frac{MD_{\overline{x}}} {n} = \frac{14.13}{34}\times = 41.57\%\\
\text{Median Coefficient of Dispersion} &= \frac{MD_{\tilde{x}}}{\tilde{x}} = \frac{13.333}{30}\times100=44.44\%
\end{align*}

Importance of Dispersion in Statistics

From the above discussion and numerical examples, In statistics, the variability or dispersion is crucial. The following are some reasons for the importance of Dispersion in Statistics:

• Understanding Data Spread: Variability gives insights into the spread or distribution of data, helping to understand how much individual data points differ from the average or some other measure.
• Data Reliability: Lower variability in data can indicate higher reliability and consistency, which is key for making sound predictions and decisions.
• Identifying Outliers: High variability can indicate the presence of outliers or anomalies in the data, which might require further investigation.
• Comparing Datasets: Dispersion measures, such as variance and standard deviation, allow for the comparison of different datasets. Two datasets might have the same mean but different levels of dispersion, which can imply different data patterns or behaviors.
• Risk Assessment: In fields like finance, assessing the variability of returns is crucial for understanding and managing risk. Higher variability often implies higher risk.
• Statistical Inferences: Many statistical methods, such as hypothesis testing and confidence intervals, rely on the variability of data to make accurate inferences about populations from samples.
• Balanced Decision Making: Understanding variability helps in making more informed decisions by providing a clearer picture of the data’s characteristics and potential fluctuations.

Overall, variability is essential for a comprehensive understanding of data, enabling analysts to draw meaningful conclusions and make informed decisions.

R Language Frequently Asked Questions

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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