Measures of Central Tendency

In statistics, a measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. It's like finding the "average" or "typical" value. There are three main measures of central tendency: the mean, the median, and the mode. Each of these measures provides a different perspective on the center of the data.

1. Mean (Arithmetic Mean)

The mean, often called the average, is the sum of all values in a dataset divided by the number of values. It is the most common measure of central tendency.

Calculation:

For a dataset $x_1, x_2, \dots, x_n$, the mean ($\bar{x}$) is calculated as:

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

Where:

• $\sum_{i=1}^{n} x_i$ is the sum of all the values in the dataset.
• $n$ is the number of values in the dataset.

Example:

Consider the following dataset of daily sales figures for a small shop: 100, 120, 110, 130, 140.

Sum of sales = 100 + 120 + 110 + 130 + 140 = 600

Number of days (n) = 5

Mean sales = $\frac{600}{5} = 120$

The average daily sale is $120.

Properties and Uses:

• It uses all the values in the dataset.
• It is sensitive to extreme values (outliers). A single very large or very small value can significantly change the mean.
• It is commonly used in many statistical calculations and tests.

2. Median

The median is the middle value in a dataset that has been ordered from least to greatest. If there is an even number of values, the median is the average of the two middle values. The median is less affected by outliers than the mean.

Calculation:

First, arrange the data in ascending order.

• If $n$ is odd, the median is the value at the position $\frac{(n+1)}{2}$.
• If $n$ is even, the median is the average of the values at positions $\frac{n}{2}$ and $(\frac{n}{2} + 1)$.

Example (Odd number of values):

Dataset: 100, 120, 110, 130, 140

Ordered dataset: 100, 110, 120, 130, 140

Number of values (n) = 5 (odd)

Median position = $\frac{(5+1)}{2} = 3$

The median is the 3rd value, which is 120.

Example (Even number of values):

Dataset: 100, 120, 110, 130, 140, 150

Ordered dataset: 100, 110, 120, 130, 140, 150

Number of values (n) = 6 (even)

Median positions = $\frac{6}{2} = 3$ and $(\frac{6}{2} + 1) = 4$

The median is the average of the 3rd and 4th values: $\frac{120 + 130}{2} = \frac{250}{2} = 125$.

Properties and Uses:

• It is not affected by extreme values.
• It is a good measure for skewed distributions.
• It is often used for income data or house prices, where there might be a few very high values.

3. Mode

The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), two modes (bimodal), or more (multimodal). If no value repeats, there is no mode.

Calculation:

Simply count the occurrences of each value and identify the one with the highest frequency.

Example (Unimodal):

Dataset: 100, 120, 110, 120, 140, 120

The value 120 appears 3 times, which is more than any other value.

Mode = 120.

Example (Bimodal):

Dataset: 100, 120, 110, 120, 140, 110

110 appears twice, and 120 appears twice.

Modes = 110 and 120.

Example (No Mode):

Dataset: 100, 120, 110, 130, 140

Each value appears only once. There is no mode.

Properties and Uses:

• It is the easiest measure to find.
• It can be used for both numerical and categorical data.
• It is not affected by extreme values.
• It may not be unique, or it may not exist.
• It is useful for identifying the most popular item or category.

Relationship between Mean, Median, and Mode

The relationship between these three measures helps in understanding the shape of a distribution.

• Symmetrical Distribution: In a perfectly symmetrical distribution (like the normal distribution), the mean, median, and mode are all equal.
• Positively Skewed Distribution (Skewed to the Right): The tail of the distribution is longer on the right side. In this case, Mode < Median < Mean. The mean is pulled towards the higher values in the tail.
• Negatively Skewed Distribution (Skewed to the Left): The tail of the distribution is longer on the left side. In this case, Mean < Median < Mode. The mean is pulled towards the lower values in the tail.

Measures of Dispersion

Measures of dispersion, also known as measures of variability or spread, describe how spread out or clustered together the values in a dataset are. While measures of central tendency tell us about the typical value, measures of dispersion tell us how consistent or varied the data is. A low dispersion indicates that the data points are close to the center, while a high dispersion indicates that they are spread out over a wider range.

1. Range

The range is the simplest measure of dispersion. It is the difference between the highest and lowest values in a dataset.

Calculation:

Range = Maximum Value - Minimum Value

Example:

Dataset: 100, 120, 110, 130, 140

Maximum Value = 140

Minimum Value = 100

Range = 140 - 100 = 40

Properties and Uses:

• Easy to calculate and understand.
• It is highly sensitive to outliers, as it only considers the two extreme values.
• It does not take into account any of the values between the maximum and minimum.
• Useful for a quick initial assessment of spread, especially in quality control charts.

2. Quartile Deviation (Semi-Interquartile Range)

The quartile deviation is based on the difference between the third quartile (Q3) and the first quartile (Q1) of the data. Quartiles divide the data into four equal parts.

• Q1 (First Quartile): The value below which 25% of the data falls.
• Q2 (Second Quartile): The median of the data (50% of data falls below it).
• Q3 (Third Quartile): The value below which 75% of the data falls.

Calculation:

First, find Q1 and Q3.

Quartile Deviation (QD) = $\frac{Q3 - Q1}{2}$

Example:

Dataset: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100

n = 10

Q1 position = $\frac{(n+1)}{4} = \frac{11}{4} = 2.75$. This means Q1 is 75% of the way between the 2nd and 3rd values. Q1 = $20 + 0.75 * (30 - 20) = 20 + 7.5 = 27.5$.

Q3 position = $\frac{3(n+1)}{4} = \frac{3(11)}{4} = \frac{33}{4} = 8.25$. This means Q3 is 25% of the way between the 8th and 9th values. Q3 = $80 + 0.25 * (90 - 80) = 80 + 2.5 = 82.5$.

QD = $\frac{82.5 - 27.5}{2} = \frac{55}{2} = 27.5$.

Note on calculating Quartiles: There are several methods for calculating quartiles, especially when $n$ is not divisible by 4. The method used above is common. Another simpler method for even $n$ is to find the median of the lower half (for Q1) and the median of the upper half (for Q3).

Properties and Uses:

• Less affected by extreme values than the range.
• Measures the spread of the middle 50% of the data.
• Useful for skewed distributions.

3. Standard Deviation

The standard deviation is the most widely used measure of dispersion. It measures the average amount of variability in a dataset. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.

Calculation:

There are two formulas: one for a population and one for a sample.

Population Standard Deviation ($\sigma$): Used when you have data for the entire population.

$$ \sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}} $$

Where:

• $\mu$ is the population mean.
• $N$ is the total number of observations in the population.

Sample Standard Deviation ($s$): Used when you have data from a sample to estimate the population standard deviation. It uses $n-1$ in the denominator (Bessel's correction) to provide a less biased estimate.

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

Where:

• $\bar{x}$ is the sample mean.
• $n$ is the number of observations in the sample.

Steps to Calculate Sample Standard Deviation:

1. Calculate the mean ($\bar{x}$) of the dataset.
2. Subtract the mean from each data point ($x_i - \bar{x}$). These are the deviations.
3. Square each deviation ($(x_i - \bar{x})^2$).
4. Sum the squared deviations ($\sum (x_i - \bar{x})^2$).
5. Divide the sum by $n-1$ (this is the variance, $s^2$).
6. Take the square root of the variance to get the standard deviation ($s$).

Example:

Dataset (sample): 10, 20, 30, 40, 50

1. Mean ($\bar{x}$) = $\frac{10+20+30+40+50}{5} = \frac{150}{5} = 30$.
2. Deviations ($x_i - \bar{x}$):
   • 10 - 30 = -20
   • 20 - 30 = -10
   • 30 - 30 = 0
   • 40 - 30 = 10
   • 50 - 30 = 20
3. Squared Deviations ($(x_i - \bar{x})^2$):
   • $(-20)^2 = 400$
   • $(-10)^2 = 100$
   • $(0)^2 = 0$
   • $(10)^2 = 100$
   • $(20)^2 = 400$
4. Sum of Squared Deviations = 400 + 100 + 0 + 100 + 400 = 1000.
5. Variance ($s^2$) = $\frac{1000}{n-1} = \frac{1000}{5-1} = \frac{1000}{4} = 250$.
6. Standard Deviation ($s$) = $\sqrt{250} \approx 15.81$.

Properties and Uses:

• It is the most reliable measure of dispersion.
• It uses all the data points.
• It is sensitive to outliers.
• It is used in many advanced statistical techniques, such as hypothesis testing and regression analysis.
• The square of the standard deviation is the variance.

4. Variance

Variance is the average of the squared differences from the mean. It is simply the square of the standard deviation.

Calculation:

Population Variance ($\sigma^2$) = $\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$

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

Properties and Uses:

• Measures the degree of variation of data points around the mean.
• Units are squared, which makes it less intuitive to interpret than standard deviation.
• Crucial in statistical theory and many applications like ANOVA (Analysis of Variance).

5. Coefficient of Variation (CV)

The coefficient of variation is a standardized measure of dispersion. It expresses the standard deviation as a percentage of the mean. It is useful for comparing the variability of two datasets with different means.

Calculation:

CV = $\frac{s}{\bar{x}} \times 100\%$ (for sample)

CV = $\frac{\sigma}{\mu} \times 100\%$ (for population)

Example:

Suppose Investment A has a mean return of 10% with a standard deviation of 5%. Investment B has a mean return of 20% with a standard deviation of 7%.

CV (A) = $\frac{5\%}{10\%} \times 100\% = 50\%$

CV (B) = $\frac{7\%}{20\%} \times 100\% = 35\%$

Although Investment B has a higher standard deviation in absolute terms, Investment A is more variable relative to its mean return. Investment B is less risky relative to its average return.

Properties and Uses:

• Unitless measure, allowing comparison across datasets with different units or scales.
• Useful when the mean is close to zero, as standard deviation alone might be misleading.
• A lower CV indicates less relative variability.

Measures of Skewness

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. In simpler terms, it tells us how much the distribution is "stretched" or "distorted" to one side compared to a symmetrical distribution (like the normal distribution). Skewness indicates the direction and extent of this asymmetry.

Understanding Skewness

A symmetrical distribution has zero skewness.

• Positive Skewness (Right Skew): The tail on the right side of the distribution is longer or fatter than the left side. The bulk of the data is concentrated on the left. In this case, the mean is typically greater than the median, which is greater than the mode (Mean > Median > Mode).
• Negative Skewness (Left Skew): The tail on the left side of the distribution is longer or fatter than the right side. The bulk of the data is concentrated on the right. In this case, the mean is typically less than the median, which is less than the mode (Mean < Median < Mode).

The magnitude of the skewness value indicates the degree of asymmetry. A value close to 0 indicates low skewness (near symmetry), while larger positive or negative values indicate greater asymmetry.

Types of Skewness Measures

There are several ways to measure skewness. The most common ones are:

1. Pearson's Coefficient of Skewness (Mode Skewness)

This measure uses the mode and standard deviation. It's particularly useful when the mode is clearly identifiable.

Calculation:

Skewness ($SK_1$) = $\frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}}$

$SK_1 = \frac{\bar{x} - \text{Mode}}{s}$

Interpretation:

• $SK_1 > 0$: Positively skewed.
• $SK_1 < 0$: Negatively skewed.
• $SK_1 = 0$: Symmetrical (or nearly so).

A common rule of thumb is:

• If $|SK_1|$ between 0 and 0.5: Approximately symmetrical.
• If $|SK_1|$ between 0.5 and 1: Moderately skewed.
• If $|SK_1|$ > 1: Highly skewed.

2. Pearson's Coefficient of Skewness (Median Skewness)

This measure uses the median and standard deviation. It's more robust than the mode-based measure because the median is less affected by extreme values. This is often preferred for moderately skewed distributions.

Calculation:

Skewness ($SK_2$) = $\frac{3 \times (\text{Mean} - \text{Median})}{\text{Standard Deviation}}$

$SK_2 = \frac{3(\bar{x} - \text{Median})}{s}$

This formula is derived from the empirical relationship between mean, median, and mode for moderately skewed distributions (Mean - Mode ≈ 3 * (Mean - Median)).

Interpretation:

• $SK_2 > 0$: Positively skewed.
• $SK_2 < 0$: Negatively skewed.
• $SK_2 = 0$: Symmetrical (or nearly so).

The same interpretation guidelines based on the magnitude apply as for $SK_1$.

3. Third Standardized Moment (Fisher-Pearson Coefficient of Skewness)

This is the most rigorous statistical measure of skewness, based on the third moment of the distribution. It is defined as the third moment about the mean divided by the cube of the standard deviation.

Calculation:

For a population: $\gamma_1 = \frac{E[(X - \mu)^3]}{\sigma^3}$

For a sample, it's estimated as: $g_1 = \frac{m_3}{s^3}$

Where:

• $E[\cdot]$ denotes the expected value.
• $\mu$ is the population mean.
• $\sigma$ is the population standard deviation.
• $m_3 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^3}{n}$ is the third sample moment about the mean.
• $s$ is the sample standard deviation.

A sample estimate that corrects for bias is:

$$ G_1 = \frac{\sqrt{n(n-1)}}{n-2} \times \frac{m_3}{s^3} $$

However, for most competitive exam purposes, the Pearson coefficients are more commonly tested.

Interpretation:

• $\gamma_1 > 0$: Positively skewed.
• $\gamma_1 < 0$: Negatively skewed.
• $\gamma_1 = 0$: Symmetrical.

4. Bowley's Coefficient of Skewness

This measure is based on quartiles and is less sensitive to extreme values than Pearson's coefficients. It is particularly useful for open-ended distributions where the mean or mode might not be calculable.

Calculation:

Bowley's Skewness ($B$) = $\frac{(Q3 + Q1) - 2 \times \text{Median}}{Q3 - Q1}$

Or equivalently:

$B = \frac{(Q3 - \text{Median}) - (\text{Median} - Q1)}{Q3 - Q1}$

Where:

• Q1 is the first quartile.
• Q3 is the third quartile.
• Median is the second quartile (Q2).

Interpretation:

• $B > 0$: Positively skewed (the upper half of the data is more spread out than the lower half).
• $B < 0$: Negatively skewed (the lower half of the data is more spread out than the upper half).
• $B = 0$: Symmetrical (the spread of the upper and lower halves relative to the median is equal).

The value ranges from -1 to +1.

Example Calculation of Skewness

Consider a dataset with the following calculated values:

• Mean ($\bar{x}$) = 50
• Median = 45
• Mode = 40
• Standard Deviation ($s$) = 15
• Q1 = 30, Q3 = 60

Calculating Pearson's Coefficients:

Using Median Skewness ($SK_2$):

$SK_2 = \frac{3 \times (50 - 45)}{15} = \frac{3 \times 5}{15} = \frac{15}{15} = 1.0$

Using Mode Skewness ($SK_1$):

$SK_1 = \frac{50 - 40}{15} = \frac{10}{15} \approx 0.67$

Both values are positive, indicating positive skewness. $SK_2 = 1.0$ suggests a high degree of skewness. The mean (50) is greater than the median (45), which is greater than the mode (40), confirming positive skew.

Calculating Bowley's Coefficient:

$B = \frac{(Q3 + Q1) - 2 \times \text{Median}}{Q3 - Q1} = \frac{(60 + 30) - 2 \times 45}{60 - 30} = \frac{90 - 90}{30} = \frac{0}{30} = 0$

Bowley's coefficient being 0 suggests symmetry in the quartile distribution. This highlights that different measures can sometimes give slightly different indications, especially if the distribution is complex or not perfectly unimodal. However, the Pearson measures are generally more sensitive to the overall shape of the distribution.