MEASURES OF DISPERSION (MEAN, VARIANCE, TYPICAL, COEFFICIENT OF VARIATION) - GROUPED DATA

Measures of Dispersion for Grouped Data
📌 Measures of dispersion quantify the separation of data points relative to the mean in a distribution.
📊 Key measures discussed include the Range (R), Mean Deviation ($D_x$), Variance ($s^2$ or $\sigma^2$), Standard Deviation ($s$ or $\sigma$), and the Coefficient of Variation (CV).
🔢 The example involves calculating these measures for the ages of 200 medical patients grouped into class intervals.

Formulas and Calculations Overview
🔍 Range (R) is calculated as the difference between the upper limit of the last interval and the lower limit of the first interval (Data Max - Data Min).
➗ Mean Deviation ($D_x$) is the arithmetic mean of the absolute differences between the class mark ($x_i$) and the arithmetic mean ($\bar{x}$), weighted by frequency ($f_i$): $\sum |x_i - \bar{x}| f_i / n$.
📈 Variance ($s^2$) for a sample uses the formula: $\sum (x_i - \bar{x})^2 f_i / (n-1)$.
💧 Standard Deviation ($s$) is the square root of the variance: $s = \sqrt{s^2}$.
🎯 Coefficient of Variation (CV) relates standard deviation to the mean: CV = (s / x) 100% (when expressed as a percentage).

Step-by-Step Example Application
⚙️ The process requires building an extended frequency table including class marks ($x_i$), frequencies ($f_i$), $x_i \cdot f_i$, $|x_i - \bar{x}|$, $|x_i - \bar{x}| \cdot f_i$, $(x_i - \bar{x})^2$, and $(x_i - \bar{x})^2 \cdot f_i$.
🧮 The mean ($\bar{x}$) was calculated as $\sum (x_i f_i) / n = 91930 / 200 = 49.65$.
📊 The calculated Range (R) was 70 (90 - 20).
➗ The Mean Deviation ($D_x$) calculated was 12.38 ($\sum |x_i - \bar{x}| f_i = 2476.5$, divided by $n=200$).
🔥 The Sample Variance ($s^2$) calculated was 227.01, using the sum of the final column ($45,175.3$) divided by $(n-1) = 199$.
📏 The Standard Deviation ($s$) obtained was 15.06 ($\sqrt{227.01}$).
💲 The Coefficient of Variation (CV) was 30.3% (15.06 / 49.65 100%).

Key Points & Insights
➡️ Using an extended frequency table is crucial to organize necessary parameters and minimize calculation errors when finding measures of dispersion for grouped data.
➡️ The Range is the simplest measure, found directly from the interval limits: Data Max - Data Min.
➡️ The Standard Deviation ($s$) indicates how spread out the data is; a smaller value means data points are closer to the mean, indicating less dispersion.
➡️ For sample calculations (like this one with $n=200$ patients), the variance denominator must use $n-1$ instead of $n$.

📸 Video summarized with SummaryTube.com on Oct 08, 2025, 21:04 UTC