1. Mean Deviation (MD)
Mean deviation measures the average absolute difference between data points and a central value (either the Mean or the Median).
| Measure |
Formula |
| MD from Mean (\(\bar{x}\)) |
\(MD_{\bar{x}} = \frac{\sum f|x - \bar{x}|}{N}\) |
| MD from Median (\(Md\)) |
\(MD_{Md} = \frac{\sum f|x - Md|}{N}\) |
| Coefficient of MD |
\(\frac{MD}{A}\) (where \(A\) is the Mean or Median used) |
Example: MD from Mean
Data: 2, 4, 6 (\(N=3\))
Calculation: \(\bar{x} = \frac{2+4+6}{3} = 4\)
| \(x\) |
\(|x - \bar{x}|\) i.e., \(|x - 4|\) |
| 2 | 2 |
| 4 | 0 |
| 6 | 2 |
| Total (\(N=3\)) |
\(\sum |x - \bar{x}| = 4\) |
Result: \(MD_{\bar{x}} = \frac{4}{3} = \mathbf{1.33}\)
Coefficient: \(\frac{1.33}{4} = \mathbf{0.33}\)
Example: MD from Median
Data: 2, 4, 6 (\(N=3\))
Calculation: \(Md = 4\) (The middle value)
| \(x\) |
\(|x - Md|\) i.e., \(|x - 4|\) |
| 2 | 2 |
| 4 | 0 |
| 6 | 2 |
| Total (\(N=3\)) |
\(\sum |x - Md| = 4\) |
Result: \(MD_{Md} = \frac{4}{3} = \mathbf{1.33}\)
Coefficient: \(\frac{1.33}{4} = \mathbf{0.33}\)
2. Quartiles and Quartile Deviation (QD)
Quartiles divide data into four equal parts. Quartile Deviation measures the dispersion of the middle 50% of the data.
• Quartile Position: \(Q_k = \text{Value of } \left[ \frac{k(N+1)}{4} \right]^{th} \text{ term}\)
• Quartile Deviation: \(QD = \frac{Q_3 - Q_1}{2}\)
• Coeff. of QD: \(\frac{Q_3 - Q_1}{Q_3 + Q_1}\)
Example: Calculating QD
Data: 10, 20, 30, 40, 50 (\(N=5\))
| Step |
Calculation |
Result |
| Find \(Q_1\) |
Position = \(\frac{1(5+1)}{4} = 1.5^{th}\) term (\(\frac{10+20}{2}\)) |
15 |
| Find \(Q_3\) |
Position = \(\frac{3(5+1)}{4} = 4.5^{th}\) term (\(\frac{40+50}{2}\)) |
45 |
| Calculate QD |
\(QD = \frac{45 - 15}{2}\) |
15 |
| Coefficient |
\(\frac{45 - 15}{45 + 15} = \frac{30}{60}\) |
0.5 |
3. Standard Deviation (\(\sigma\))
Standard deviation shows how much the data points deviate from the mean. It is the most reliable measure of dispersion.
| Series Type |
Formula |
| Individual |
\(\sigma = \sqrt{\frac{\sum x^2}{N} - (\bar{x})^2}\) |
| Discrete |
\(\sigma = \sqrt{\frac{\sum fx^2}{N} - \left(\frac{\sum fx}{N}\right)^2}\) |
Example: Discrete Series
Data: \(x = \{2, 4\}\), \(f = \{1, 3\}\) (\(N=4\))
| \(x\) |
\(f\) |
\(fx\) |
\(fx^2\) |
| 2 |
1 |
2 |
4 |
| 4 |
3 |
12 |
48 |
| Total |
\(N = 4\) |
\(\sum fx = 14\) |
\(\sum fx^2 = 52\) |
Step 1: Mean (\(\bar{x}\)) = \(\frac{14}{4} = 3.5\)
Step 2: \(\sigma = \sqrt{\frac{52}{4} - (3.5)^2} = \sqrt{13 - 12.25} = \sqrt{0.75} = \mathbf{0.866}\)
4. Box and Whisker Plot
This graphical representation is built using the 5-Number Summary derived from the partition values.
| Feature |
Definition |
Data Example (10, 20, 30, 40, 50) |
| Minimum |
The lowest value in the dataset. |
10 (Start of lower whisker) |
| \(Q_1\) |
The First Quartile (25th percentile). |
15 (Start of the box) |
| Median (\(Md\)) |
The middle value (50th percentile). |
30 (Line inside the box) |
| \(Q_3\) |
The Third Quartile (75th percentile). |
45 (End of the box) |
| Maximum |
The highest value in the dataset. |
50 (End of upper whisker) |
Note: The "Box" represents the Interquartile Range (IQR), encompassing the middle 50% of the data from \(Q_1\) to \(Q_3\).