9.1 Mean, Median, Mode, Range, Mean Deviation & Standard Deviation

Measures of Central Tendency

Measures of central tendency identify a single representative central value around which numerical observations cluster.

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|                       MEASURES OF CENTRAL TENDENCY                          |
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| ARITHMETIC MEAN (x̄): Mathematical centroid; sensitive to extreme outliers.  |
| MEDIAN (M):          Positional midpoint; less sensitive to outlier magnitude.     |
| MODE (Z):            Point of maximum frequency density; practical peak.    |
| EMPIRICAL RULE:      Mode approximately 3(Median) - 2(Mean) [empirical estimate]    |
+-----------------------------------------------------------------------------+

1. Arithmetic Mean ($\bar{x}$)

The arithmetic mean is the quotient of the sum of all observations divided by the total number of observations.

• Raw (Ungrouped) Data:

$$\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$$

• Discrete Frequency Distribution:

$$\bar{x} = \frac{\sum_{i=1}^k f_i x_i}{\sum_{i=1}^k f_i} = \frac{\sum f_i x_i}{N} \quad (\text{where } N = \sum f_i)$$

• Grouped Continuous Frequency Distribution: In grouped data, each class interval is represented by its class mark (midpoint) $x_i = \frac{\text{Lower limit} + \text{Upper limit}}{2}$. Three methods streamline calculation:

• Combined Mean Formula: If two groups contain $n_1$ and $n_2$ observations with respective means $\bar{x}_1$ and $\bar{x}_2$, their pooled composite mean is:

$$\bar{x}_{12} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}$$

2. Median ($M$)

The median is a central value of ordered data: at least half the observations are no greater than it and at least half are no less than it. With ties, there need not be exactly half strictly below or above it.

• Ungrouped Data: Arrange $n$ values in ascending order:
  • If $n$ is odd: $\text{Median} = \left(\frac{n + 1}{2}\right)\text{-th observation}$.
  • If $n$ is even: $\text{Median} = \frac{1}{2} \left[ \left(\frac{n}{2}\right)\text{-th} + \left(\frac{n}{2} + 1\right)\text{-th observation} \right]$.
• Grouped Continuous Distribution:
  1. Construct the cumulative frequency ($CF$) distribution.
  2. Determine $\frac{N}{2}$, where $N = \sum f_i$.
  3. Locate the median class: the class interval whose cumulative frequency just exceeds or equals $\frac{N}{2}$.
  4. Apply the interpolation formula:

$$M = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times h$$

Where:

• $L = \text{Lower limit of the median class}$
• $N = \text{Total frequency } (\sum f_i)$
• $CF = \text{Cumulative frequency of the class immediately PRECEDING the median class}$
• $f = \text{Simple frequency of the median class itself}$
• $h = \text{Class width of the median class}$

3. Mode ($Z$)

The mode is the value that occurs with the highest frequency in a dataset.

• Grouped Continuous Distribution:
  1. Locate the modal class: the class interval possessing the highest frequency ($f_1$).
  2. Apply the modal formula:

$$\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$$

Where:

• $L = \text{Lower limit of the modal class}$
• $f_1 = \text{Frequency of the modal class}$
• $f_0 = \text{Frequency of the class immediately PRECEDING the modal class}$
• $f_2 = \text{Frequency of the class immediately SUCCEEDING the modal class}$
• $h = \text{Class width of the modal class}$

4. Karl Pearson's Empirical Skewness Relationship

For moderately asymmetrical (unimodal) frequency distributions, the three central tendencies can approximately satisfy the empirical relationship:

$$\text{Mode} \approx 3\text{Median} - 2\text{Mean}$$

Alternative algebraic expressions useful for rapid CBT substitutions:

• $\text{Mean} - \text{Mode} \approx 3(\text{Mean} - \text{Median})$
• $\text{Median} \approx \frac{1}{3}\text{Mode} + \frac{2}{3}\text{Mean}$

Symmetric unimodal examples:    Mean = Median = Mode
Typical unimodal right-tail examples: Mean > Median > Mode (not universal)
Typical unimodal left-tail examples: Mode > Median > Mean (not universal)

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Measures of Dispersion (Spread)

Central tendency describes the location of data; dispersion measures the degree of scatter or variability of individual values around the central point.

1. Range & Coefficient of Range

• Absolute Range: Difference between extreme values:

$$\text{Range} = X_{\max} - X_{\min}$$

• Coefficient of Range (Relative):

$$\text{Coefficient of Range} = \frac{X_{\max} - X_{\min}}{X_{\max} + X_{\min}}$$

2. Mean Deviation ($MD$)

The arithmetic average of the absolute deviations taken from a central value (mean or median):

• About Mean ($\bar{x}$):

$$MD(\bar{x}) = \frac{\sum_{i=1}^n |x_i - \bar{x}|}{n} = \frac{\sum f_i |x_i - \bar{x}|}{N}$$

• About Median ($M$):

$$MD(M) = \frac{\sum_{i=1}^n |x_i - M|}{n} = \frac{\sum f_i |x_i - M|}{N}$$

3. Variance ($\sigma^2$) and Standard Deviation ($\sigma$)

Standard deviation is the nonnegative square root of the arithmetic mean of squared deviations taken from the arithmetic mean (the second central moment).

• Variance Formula for Raw Data:

$$\sigma^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - (\bar{x})^2$$

• Grouped Continuous Data (Shortcut Method):

$$\sigma = \sqrt{ \frac{\sum f_i d_i^2}{N} - \left( \frac{\sum f_i d_i}{N} \right)^2 } \times h \quad \left( \text{where } d_i = \frac{x_i - A}{h} \right)$$

• Standard Deviation ($\sigma$):

$$\sigma = \sqrt{\sigma^2}$$

4. Coefficient of Variation ($CV$)

Developed by Karl Pearson, the Coefficient of Variation expresses dispersion as a percentage of the mean:

$$CV = \frac{\sigma}{\bar{x}} \times 100\%$$

• Consistency Rule: When comparing two series (e.g., punctuality of two railway divisions):
  • Lower $CV \implies$ Higher consistency, greater uniformity, more stable operation.
  • Higher $CV \implies$ Greater variability, less consistency.

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Mathematical Transformations: Effects of Shifting & Scaling

Examinees frequently encounter questions where every observation in a dataset undergoes a linear algebraic transformation $y_i = a \cdot x_i + b$, where $b$ represents a change of origin (shift) and $a$ represents a change of scale (multiplication).

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Worked Examples

Worked Example 1: Grouped Data Median and Mode of Sectional Running Speeds

Problem: A Section Controller tabulates the recorded operational speeds (in km/h) of 50 freight trains traversing a critical junction section:

Calculate: (a) The Median speed, and (b) The Modal speed.

• Step 1: Compute Cumulative Frequencies ($CF$):
  • $10-20: f = 5, CF = 5$
  • $20-30: f = 8, CF = 5 + 8 = 13$
  • $30-40: f = 15, CF = 13 + 15 = 28$
  • $40-50: f = 12, CF = 28 + 12 = 40$
  • $50-60: f = 10, CF = 40 + 10 = 50$ Total frequency $N = 50$.
• Step 2: Calculate Median ($M$):
  • Half-count: $\frac{N}{2} = \frac{50}{2} = 25$.
  • The cumulative frequency just exceeding 25 is 28, corresponding to the class 30 – 40.
  • Parameter values: $L = 30, CF = 13, f = 15, h = 10$.
  • Substitute into the median interpolation formula:

$$M = L + \left( \frac{\frac{N}{2} - CF}{f} \right) h = 30 + \left( \frac{25 - 13}{15} \right) \times 10 = 30 + \left( \frac{12}{15} \right) \times 10$$ $$M = 30 + 0.8 \times 10 = 30 + 8 = 38.00 \text{ km/h}$$

• Step 3: Calculate Mode ($Z$):
  • Highest frequency is $f_1 = 15$, situated in class 30 – 40.
  • Preceding frequency $f_0 = 8$; Succeeding frequency $f_2 = 12$; Lower limit $L = 30$; Class width $h = 10$.
  • Substitute into the mode formula:

$$\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) h = 30 + \left( \frac{15 - 8}{2(15) - 8 - 12} \right) \times 10$$ $$\text{Mode} = 30 + \left( \frac{7}{30 - 20} \right) \times 10 = 30 + \left( \frac{7}{10} \right) \times 10 = 30 + 7 = 37.00 \text{ km/h}$$

• Conclusion: The grouped-data estimates are median 38.00 km/h and mode 37.00 km/h; they do not recover the exact original observations.

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Worked Example 2: Karl Pearson's Empirical Rule and Composite Means

Problem: In one fictional running-time dataset, the mean running time is $54.00$ minutes and the median running time is $51.00$ minutes. Using Pearson's empirical formula, estimate the mode. In a separate pooling exercise, if Division X ($n_1 = 30$ trains, $\bar{x}_1 = 60$ minutes) merges operations with Division Y ($n_2 = 20$ trains, $\bar{x}_2 = 75$ minutes), find the aggregate combined mean.

• Step 1: Estimate Mode using Pearson's empirical formula:

$$\text{Mode} \approx 3\text{Median} - 2\text{Mean}$$ $$\text{Mode} = 3(51.00) - 2(54.00) = 153.00 - 108.00 = 45.00 \text{ minutes}$$

• Step 2: Calculate Combined Mean for merged divisions:

$$\bar{x}_{12} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2} = \frac{30(60) + 20(75)}{30 + 20}$$ $$\bar{x}_{12} = \frac{1800 + 1500}{50} = \frac{3300}{50} = 66.00 \text{ minutes}$$

• Conclusion: The estimated mode is 45.00 minutes and the combined mean is 66.00 minutes.

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Worked Example 3: Effect of Linear Transformation on Variance and Standard Deviation

Problem: A sample of 12 block section track occupancy times has a mean $\bar{x} = 25.0$ minutes, a standard deviation $\sigma = 4.0$ minutes, and a variance $\sigma^2 = 16.0 \text{ minutes}^2$. For a mathematical exercise, transform each recorded value by the algebraic rule:

$$y_i = 3x_i - 10$$

Determine the new mean ($\bar{y}$), new standard deviation ($\sigma_y$), new variance ($\sigma_y^2$), and the new coefficient of variation ($CV_y$).

• Step 1: Compute New Mean ($\bar{y}$):

$$\bar{y} = 3\bar{x} - 10 = 3(25.0) - 10 = 75.0 - 10 = 65.0 \text{ minutes}$$

• Step 2: Compute New Standard Deviation ($\sigma_y$): Origin shift ($-10$) has zero effect on dispersion. Scale factor is $a = 3$.

$$\sigma_y = |a| \times \sigma_x = 3 \times 4.0 = 12.0 \text{ minutes}$$

• Step 3: Compute New Variance ($\sigma_y^2$):

$$\sigma_y^2 = a^2 \times \sigma_x^2 = 3^2 \times 16.0 = 9 \times 16.0 = 144.0 \text{ minutes}^2$$

(Check: $\sigma_y^2 = 12.0^2 = 144.0$.)

• Step 4: Compute New Coefficient of Variation ($CV_y$):

$$CV_y = \frac{\sigma_y}{\bar{y}} \times 100\% = \frac{12.0}{65.0} \times 100\% = \frac{1200}{65} \approx 18.46\%$$

• Conclusion: $\bar{y} = 65.0 \text{ min}$, $\sigma_y = 12.0 \text{ min}$, $\sigma_y^2 = 144.0 \text{ min}^2$, and $CV_y \approx 18.46\%$.

Grouped mean and median/mode interpolation are estimates based on class information, not the exact original observations. The displayed grouped mode formula assumes equal-width continuous classes and an appropriate modal class. Pearson’s empirical relation and familiar mean–median–mode orderings are approximations for suitable unimodal distributions, not universal identities. CV comparisons require a meaningful positive ratio-scale mean; a zero or negative mean makes the usual interpretation unsuitable. The displayed variance divides by N for the described population; an unbiased sample variance uses n - 1.