Dispersion – Range, Quartile Deviation, Standard Deviation
In statistics, dispersion refers to the spread or variability of a set of data. It tells us how much the individual data points differ from each other and from the central tendency (like the mean or median) of the dataset. Understanding dispersion is crucial because two datasets can have the same mean but vastly different spreads, indicating different underlying patterns or distributions of the data. For instance, two classes might have the same average score on a test, but one class might have scores clustered tightly around the average, while the other has scores spread out widely. Dispersion measures help us quantify this spread.
1. Range
The range is the simplest and most straightforward measure of dispersion. It is defined as the difference between the highest (maximum) value and the lowest (minimum) value in a dataset.
Formula for Range:
Range = Maximum Value – Minimum Value
Explanation:
To calculate the range, you simply identify the largest number and the smallest number in your data set and subtract the smallest from the largest. The result is the range.
Example:
Consider the following set of scores for a group of students on a quiz: 5, 8, 12, 3, 10, 7, 15.
- The Maximum Value is 15.
- The Minimum Value is 3.
- Range = 15 – 3 = 12.
So, the range of these scores is 12.
Advantages of Range:
- It is very easy to calculate and understand.
- It gives a quick idea of the spread of the data.
Disadvantages of Range:
- It is highly affected by extreme values (outliers). A single very high or very low score can drastically change the range, making it not a robust measure.
- It only considers the two extreme values and ignores all other data points.
- It is not suitable for open-ended frequency distributions where the minimum or maximum values are not known.
2. Quartile Deviation (QD)
Quartile Deviation, also known as Semi-Interquartile Range, is a measure of dispersion that is less affected by extreme values compared to the range. It is calculated based on the quartiles of the data. Quartiles divide a dataset into four equal parts.
Understanding Quartiles:
When a dataset is arranged in ascending order:
- Q1 (First Quartile or Lower Quartile): The value below which 25% of the data falls. It is the median of the lower half of the data.
- Q2 (Second Quartile): This is the median of the entire dataset. 50% of the data falls below Q2.
- Q3 (Third Quartile or Upper Quartile): The value below which 75% of the data falls. It is the median of the upper half of the data.
Formula for Quartile Deviation:
Quartile Deviation (QD) = (Q3 – Q1) / 2
The term (Q3 – Q1) is called the Interquartile Range (IQR). So, QD is half of the IQR.
Steps to Calculate Quartile Deviation:
- Arrange the data in ascending order.
- Find the Median (Q2).
- Find the First Quartile (Q1): This is the median of the data points below the Median (Q2).
- Find the Third Quartile (Q3): This is the median of the data points above the Median (Q2).
- Calculate the Interquartile Range (IQR) = Q3 – Q1.
- Calculate Quartile Deviation (QD) = IQR / 2.
Example (Ungrouped Data):
Consider the dataset: 2, 4, 6, 8, 10, 12, 14, 16, 18.
- The data is already in ascending order.
- Total number of observations (n) = 9.
- Median (Q2) = The (9+1)/2 = 5th observation, which is 10.
- Lower half of the data (excluding median if n is odd): 2, 4, 6, 8.
- Q1 = Median of the lower half = (4 + 6) / 2 = 5.
- Upper half of the data (excluding median if n is odd): 12, 14, 16, 18.
- Q3 = Median of the upper half = (14 + 16) / 2 = 15.
- IQR = Q3 – Q1 = 15 – 5 = 10.
- QD = IQR / 2 = 10 / 2 = 5.
Example (Grouped Data - Frequency Distribution):
For grouped data, quartiles are calculated using the formula: Qk = L + [ ( (nk/4) - CF ) / f ] * i Where:
- k = 1 for Q1, 3 for Q3
- L = Lower boundary of the quartile class
- n = Total frequency
- CF = Cumulative frequency of the class preceding the quartile class
- f = Frequency of the quartile class
- i = Class interval
Let's assume a frequency table:
| Class Interval | Frequency (f) | Cumulative Frequency (CF) |
|---|---|---|
| 0-10 | 5 | 5 |
| 10-20 | 8 | 13 |
| 20-30 | 15 | 28 |
| 30-40 | 10 | 38 |
| 40-50 | 7 | 45 |
Total frequency (n) = 45.
Calculating Q1: n/4 = 45/4 = 11.25. The quartile class is the one where CF is just greater than 11.25, which is 20-30 (CF=28). Q1 = 20 + [ (11.25 - 13) / 15 ] * 10 = 20 + [ -1.75 / 15 ] * 10 = 20 - 1.17 = 18.83 (approximately).
Calculating Q3: 3n/4 = 3 * 45 / 4 = 135 / 4 = 33.75. The quartile class is the one where CF is just greater than 33.75, which is 30-40 (CF=38). Q3 = 30 + [ (33.75 - 28) / 10 ] * 10 = 30 + [ 5.75 / 10 ] * 10 = 30 + 5.75 = 35.75.
IQR = Q3 – Q1 = 35.75 – 18.83 = 16.92. QD = IQR / 2 = 16.92 / 2 = 8.46.
Advantages of Quartile Deviation:
- It is not affected by extreme values (outliers) as it only uses the middle 50% of the data.
- It can be calculated for open-ended distributions if the median and quartiles fall within the closed intervals.
Disadvantages of Quartile Deviation:
- It does not take into account all the observations in the dataset.
- It is not as commonly used or as statistically powerful as standard deviation.
- It is a less stable measure of dispersion compared to standard deviation, especially for small samples.
3. Standard Deviation (SD)
Standard Deviation is the most widely used and statistically robust measure of dispersion. It measures the average amount by which each data point in a dataset deviates from the mean. A low standard deviation indicates that the data points are clustered closely around the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.
Understanding Variance:
Standard deviation is the square root of the variance. Variance is the average of the squared differences from the mean. Squaring the differences ensures that all values are positive and gives more weight to larger deviations.
Formulas for Standard Deviation:
For a Population (σ): σ = √[ Σ(xi – μ)² / N ] Where:
- σ (sigma) is the population standard deviation.
- xi is each individual data point.
- μ (mu) is the population mean.
- N is the total number of observations in the population.
- Σ denotes the sum of.
For a Sample (s): s = √[ Σ(xi – x̄)² / (n-1) ] Where:
- s is the sample standard deviation.
- xi is each individual data point in the sample.
- x̄ (x-bar) is the sample mean.
- n is the total number of observations in the sample.
- (n-1) is used for Bessel's correction, which provides a less biased estimate of the population variance when using a sample.
Steps to Calculate Standard Deviation (for a Sample):
- Calculate the sample mean (x̄).
- For each data point (xi), find the difference between the data point and the mean (xi – x̄).
- Square each of these differences: (xi – x̄)².
- Sum up all the squared differences: Σ(xi – x̄)².
- Divide the sum of squared differences by (n-1) to get the sample variance (s²).
- Take the square root of the variance to get the sample standard deviation (s).
Example (Ungrouped Data - Sample):
Consider the sample dataset: 5, 8, 12, 3, 10, 7, 15. (n = 7)
- Calculate Mean (x̄): x̄ = (5 + 8 + 12 + 3 + 10 + 7 + 15) / 7 = 60 / 7 ≈ 8.57
- Calculate Deviations from Mean (xi – x̄): 5 – 8.57 = -3.57 8 – 8.57 = -0.57 12 – 8.57 = 3.43 3 – 8.57 = -5.57 10 – 8.57 = 1.43 7 – 8.57 = -1.57 15 – 8.57 = 6.43
- Square the Deviations (xi – x̄)²: (-3.57)² ≈ 12.74 (-0.57)² ≈ 0.32 (3.43)² ≈ 11.76 (-5.57)² ≈ 31.02 (1.43)² ≈ 2.04 (-1.57)² ≈ 2.46 (6.43)² ≈ 41.35
- Sum of Squared Deviations Σ(xi – x̄)²: 12.74 + 0.32 + 11.76 + 31.02 + 2.04 + 2.46 + 41.35 ≈ 101.69
- Calculate Variance (s²): s² = Σ(xi – x̄)² / (n-1) = 101.69 / (7-1) = 101.69 / 6 ≈ 16.95
- Calculate Standard Deviation (s): s = √s² = √16.95 ≈ 4.12
The standard deviation of this sample is approximately 4.12.
Example (Grouped Data - Sample):
For grouped data, the formula for sample standard deviation is: s = √[ Σf(xi – x̄)² / (n-1) ] Where:
- f is the frequency of each class.
- xi is the midpoint of each class.
- x̄ is the sample mean.
- n is the total frequency.
Let's use the previous frequency table, assuming it represents a sample.
| Class Interval | Midpoint (xi) | Frequency (f) | f * xi | (xi – x̄) | (xi – x̄)² | f * (xi – x̄)² |
|---|---|---|---|---|---|---|
| 0-10 | 5 | 5 | 25 | (5 - 30.67) = -25.67 | 658.95 | 3294.75 |
| 10-20 | 15 | 8 | 120 | (15 - 30.67) = -15.67 | 245.55 | 1964.40 |
| 20-30 | 25 | 15 | 375 | (25 - 30.67) = -5.67 | 32.15 | 482.25 |
| 30-40 | 35 | 10 | 350 | (35 - 30.67) = 4.33 | 18.75 | 187.50 |
| 40-50 | 45 | 7 | 315 | (45 - 30.67) = 14.33 | 205.35 | 1437.45 |
| Total | n = 45 | 1185 | 7366.35 |
Calculate Mean (x̄): x̄ = Σ(f * xi) / n = 1185 / 45 = 26.33 (using midpoint approximation for mean) *Correction*: Let's re-calculate the mean and deviations with a more precise mean. x̄ = 1185 / 45 = 26.333...
Let's recalculate the deviations and squared deviations with the precise mean (26.333...): Class Interval | Midpoint (xi) | Frequency (f) | f * xi | (xi - x̄) | (xi - x̄)² | f * (xi - x̄)² ---|---|---|---|---|---|--- 0-10 | 5 | 5 | 25 | (5 - 26.33) = -21.33 | 455.00 | 2275.00 10-20 | 15 | 8 | 120 | (15 - 26.33) = -11.33 | 128.37 | 1026.96 20-30 | 25 | 15 | 375 | (25 - 26.33) = -1.33 | 1.77 | 26.55 30-40 | 35 | 10 | 350 | (35 - 26.33) = 8.67 | 75.17 | 751.70 40-50 | 45 | 7 | 315 | (45 - 26.33) = 18.67 | 348.57 | 2439.99 Total | | n = 45 | 1185 | | | 6510.20
Calculate Variance (s²): s² = Σf(xi – x̄)² / (n-1) = 6510.20 / (45-1) = 6510.20 / 44 ≈ 147.96
Calculate Standard Deviation (s): s = √s² = √147.96 ≈ 12.16
The sample standard deviation for this grouped data is approximately 12.16.
Interpretation of Standard Deviation:
Standard deviation gives us a measure of the typical deviation from the mean. For example, if the mean height of a group of plants is 20 cm and the standard deviation is 2 cm, it means that most plants will have heights around 20 cm, with typical heights ranging from 18 cm (20-2) to 22 cm (20+2).
Advantages of Standard Deviation:
- It is the most reliable measure of dispersion as it uses all data points.
- It is less affected by sampling fluctuations than other measures.
- It is used in many advanced statistical formulas and tests (e.g., t-tests, ANOVA).
- It has a clear algebraic definition and is amenable to further statistical treatment.
Disadvantages of Standard Deviation:
- It is more complex to calculate than range or quartile deviation, especially for ungrouped data.
- It is affected by extreme values (outliers), though less so than the range.
- It cannot be calculated for open-ended frequency distributions.
- The interpretation is relative; it is best understood when compared with the mean or other standard deviations.
Comparing Measures of Dispersion
Each measure of dispersion has its own strengths and weaknesses, making them suitable for different situations.
| Measure | Formula | Sensitivity to Outliers | Data Requirement | Usefulness |
|---|---|---|---|---|
| Range | Max - Min | High | All values | Quick estimate, simple cases |
| Quartile Deviation (QD) | (Q3 - Q1) / 2 | Low | All values (for quartiles) | Robust measure, open-ended distributions (if median/quartiles are within bounds) |
| Standard Deviation (SD) | √[ Σ(xi – x̄)² / (n-1) ] (sample) | Moderate | All values | Most common, statistically powerful, widely used in inference |
Coefficient of Variation (CV)
While not a direct measure of dispersion itself, the Coefficient of Variation is a useful statistic derived from the mean and standard deviation. It expresses the standard deviation as a percentage of the mean. This allows for comparison of the relative variability between two datasets with different means.
Formula: CV = (s / x̄) * 100% (for a sample) CV = (σ / μ) * 100% (for a population)
A higher CV indicates greater relative variability.
Example:
Dataset A: Mean = 50, SD = 10. CV = (10 / 50) * 100% = 20%. Dataset B: Mean = 100, SD = 15. CV = (15 / 100) * 100% = 15%.
Although Dataset B has a larger standard deviation (15 vs 10), Dataset A has higher relative variability (20% vs 15%) because its standard deviation is a larger proportion of its mean.