Dispersion Calculator

Range:

Interquartile Range (IQR):

Quartile Deviation:

Sample Variance:

Sample Standard Deviation:

1. What is the Dispersion Calculator?

Definition: The Dispersion Calculator computes measures of dispersion for a dataset, including range, interquartile range (IQR), quartile deviation, sample variance, and sample standard deviation, to describe data spread.

Purpose: This tool is used in statistics to quantify the variability or spread of data, aiding in data analysis for research, finance, and quality control.

2. How Does the Calculator Work?

The calculator uses the following formulas:

\( \text{Range} = \max - \min \)

\( \text{IQR} = Q_3 - Q_1 \)

\( \text{Quartile Deviation} = \frac{\text{IQR}}{2} \)

\( s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1} \quad \text{(Sample Variance)} \)

\( s = \sqrt{s^2} \quad \text{(Sample Standard Deviation)} \)

Where:

\( \max, \min \): Maximum and minimum values;
\( Q_1, Q_3 \): First and third quartiles;
\( x_i \): Data points;
\( \bar{x} \): Mean;
\( n \): Number of data points;
\( s^2, s \): Sample variance and standard deviation.

Steps:

Enter a comma-separated list of numbers (at least 2 for range/variance, 4 for IQR/quartile deviation).
Calculate range as maximum minus minimum.
Sort data to compute quartiles (Q1, Q3), then calculate IQR and quartile deviation.
Compute mean, then calculate sample variance and standard deviation.
Display results formatted to four decimal places or scientific notation, with N/A for IQR and quartile deviation if fewer than 4 numbers.

3. Importance of Dispersion Measures

Dispersion measures are essential for:

Data Spread Analysis: Quantify how spread out or clustered data points are.
Statistical Insights: Support data comparison in research, finance, and quality control.
Outlier Detection: IQR helps identify outliers for robust data analysis.

4. Using the Calculator

Example: Calculate dispersion for the dataset: [4, 7, 2, 8, 7, 10].

Input: 4,7,2,8,7,10
Sorted: [2, 4, 7, 7, 8, 10]
Range: \( 10 - 2 = 8 \)
Mean: \( \frac{4+7+2+8+7+10}{6} \approx 6.3333 \)
Variance: \( \frac{(4-6.3333)^2 + \cdots + (10-6.3333)^2}{6-1} \approx 8.6667 \)
Std Dev: \( \sqrt{8.6667} \approx 2.9439 \)
IQR: Q1 = median([2, 4]) = 3, Q3 = median([7, 8, 10]) = 8, IQR = \( 8 - 3 = 5 \)
Quartile Deviation: \( \frac{5}{2} = 2.5 \)
Result: Range: 8.0000, IQR: 5.0000, Quartile Deviation: 2.5000, Variance: 8.6667, Std Dev: 2.9439

5. Frequently Asked Questions (FAQ)

Q: What are dispersion measures?
A: Dispersion measures (range, IQR, quartile deviation, variance, standard deviation) quantify the spread or variability of a dataset.

Q: Why use sample variance instead of population variance?
A: Sample variance uses \( n-1 \) to correct for bias when estimating population variance from a sample.

Q: Why does IQR require at least 4 numbers?
A: IQR needs enough data points to define meaningful lower and upper halves for quartile calculations.