Statistical Methods in Economics | CUET PG Economics – Notes

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1. Mean, Median & Mode

1.1. Central Tendency and Mathematical Averages

1.2. Positional Averages and Partition Values

1.3. Mode and Distributional Relationships

1.4. Comparative Properties and Economic Suitability

2. Skewness

3. Quartile Deviation, Average Deviation and Standard Deviation

3.1. Quartile Deviation

3.2. Average Deviation

3.3. Standard Deviation

3.4. Relationship Between the Measures

4. Correlation

4.1. Conceptual Foundations and Classification

4.2. Karl Pearson’s Coefficient of Correlation

4.3. Spearman’s Rank Correlation and Concurrent Deviation

5. Probability Distribution Sampling

5.1. Introduction

5.2. Basic Terminology

5.3. Types of Sampling

5.4. Sampling Distribution

5.5. Other Important Sampling Distributions

5.6. Properties of a Good Estimator

5.7. Key Formulas

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Mean, Median & Mode

Central Tendency and Mathematical Averages

Central tendency represents the single central value around which an entire dataset clusters. According to George Udny Yule’s classic criteria, an ideal measure of central tendency must be rigidly defined, based on all observations, readily understandable, simple to calculate, amenable to further algebraic treatment, and minimally affected by sampling fluctuations. Measures of location are broadly classified into mathematical averages (Arithmetic Mean, Geometric Mean, Harmonic Mean) and positional averages (Median, Mode).

The Arithmetic Mean (\(\bar{x}\)) is the most fundamental mathematical measure. For an unweighted series of $n$ observations \(x_1, x_2, \dots, x_n\), it is defined as:

$$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$$
For grouped frequency distributions with frequencies $f_i$ corresponding to class midpoints \(x_i\) where \(N = \sum f_i\):

$$\bar{x} = \frac{\sum_{i=1}^{k} f_i x_i}{N}$$
The arithmetic mean satisfies critical mathematical properties essential for econometric modeling. First, the sum of algebraic deviations of individual observations from their arithmetic mean is strictly equal to zero:

$$\sum_{i=1}^{n} (x_i – \bar{x}) = 0$$
Second, the sum of squared deviations from the arithmetic mean is a minimum compared to deviations taken from any other arbitrary origin \(A\):

$$\sum_{i=1}^{n} (x_i – \bar{x})^2 < \sum_{i=1}^{n} (x_i – A)^2 \quad \forall A \neq \bar{x}$$
Third, the arithmetic mean is invariant under a change of origin and scale. If a new variable \(y\) is defined as \(y_i = a + b x_i\), where \(a\) is the shift in origin and $b$ is the change in scale, the new mean is given by:

$$\bar{y} = a + b \bar{x}$$
Fourth, if a dataset is partitioned into $k$ sub-groups with sample sizes \(n_1, n_2, \dots, n_k\) and respective sub-means \(\bar{x}_1, \bar{x}_2, \dots, \bar{x}_k\), the combined or composite mean \(\bar{x}_c\) is calculated as:

$$\bar{x}_c = \frac{\sum_{i=1}^{k} n_i \bar{x}_i}{\sum_{i=1}^{k} n_i}$$
When observations carry varying relative weights \(w_i\), the weighted arithmetic mean is evaluated as:

$$\bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}$$
The Geometric Mean (\(GM\)) is defined as the \(n\)-th root of the product of $n$ positive observations:

$$GM = \left( \prod_{i=1}^{n} x_i \right)^{\frac{1}{n}} = \text{antilog}\left( \frac{1}{n} \sum_{i=1}^{n} \log x_i \right)$$
For grouped continuous data, \(GM = \text{antilog}\left( \frac{1}{N} \sum f_i \log x_i \right)\). The geometric mean is undefined if any observation is zero or negative. It is specifically suited for measuring relative changes, percentage growth rates, financial compound interest, index number construction (satisfying the time-reversal and factor-reversal tests in Fisher’s ideal index), and ratios.

The Harmonic Mean (\(HM\)) is the reciprocal of the arithmetic mean of the reciprocals of individual non-zero observations:

$$HM = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$$
For continuous grouped frequency data, \(HM = \frac{N}{\sum \left( \frac{f_i}{x_i} \right)}\). Harmonic mean is preferred when dealing with rates, speed, ratios, or unit-value variables where time or distance remains constant.

For any strictly positive dataset, the mathematical averages adhere to the rigid algebraic inequality:

$$AM \ge GM \ge HM$$
The equality holds if and only if all individual observations in the dataset are strictly equal (\(x_1 = x_2 = \dots = x_n\)). Furthermore, for any two positive numbers, the geometric mean equals the geometric square root of the product of the arithmetic and harmonic means:

$$GM^2 = AM \times HM$$

Positional Averages and Partition Values

The Median (\(M_e\)) is the value of the middlemost item when the dataset is arranged in ascending or descending order of magnitude. It divides the total frequency into two equal halves such that $50\%$ of observations lie below it and $50\%$ lie above it. For an ungrouped dataset of $n$ observations arranged in order, the median position is:

$$\text{Position of } M_e = \frac{n + 1}{2} \text{ th item (for odd } n \text{)}$$
When \(n\) is even, the median is the arithmetic mean of the \(\left(\frac{n}{2}\right)\)-th and \(\left(\frac{n}{2} + 1\right)\)-th observations. For a continuous grouped frequency distribution, the median class is identified using the cumulative frequency corresponding to \(\frac{N}{2}\), and the exact value is derived via linear interpolation:

$$M_e = L + \left( \frac{\frac{N}{2} – C}{f} \right) \times h$$
Where \(L\) is the lower limit of the median class, \(N\) is the total frequency, \(C\) is the cumulative frequency of the pre-median class, \(f\) is the simple frequency of the median class, and \(h\) is the class interval length.

The median possesses a distinct mathematical property regarding absolute deviation minimization: the sum of absolute deviations of observations from a point is minimum when that point is the median:

$$\sum_{i=1}^{n} \vert{}x_i – M_e\vert{} \le \sum_{i=1}^{n} \vert{}x_i – A\vert{} \quad \forall A$$
Graphically, the median is located at the abscissa of the intersection point of the less-than ogive and more-than ogive curves.

The median is a specific case of general partition values or quantiles. Quartiles split ordered data into four equal parts using three cut-off points (\(Q_1, Q_2, Q_3\)). For continuous distributions:

$$Q_1 = L + \left( \frac{\frac{N}{4} – C}{f} \right) \times h, \quad Q_3 = L + \left( \frac{\frac{3N}{4} – C}{f} \right) \times h$$
Here, \(Q_2\) is identical to the median \(M_e\). Deciles partition data into ten equal parts using nine values (\(D_1, \dots, D_9\)), where \(D_k = L + \left( \frac{\frac{kN}{10} – C}{f} \right) \times h\). Percentiles divide data into one hundred equal parts using ninety-nine values (\(P_1, \dots, P_{99}\)), where \(P_k = L + \left( \frac{\frac{kN}{100} – C}{f} \right) \times h\). The 50th percentile (\(P_{50}\)), 5th decile (\(D_5\)), and 2nd quartile (\(Q_2\)) are all mathematically equivalent to the median.

Mode and Distributional Relationships

The Mode (\(M_o\)) is the observation that occurs with the maximum frequency in a distribution. It represents the point of maximum concentration of values. A distribution with one modal peak is unimodal, two peaks is bimodal, and multiple peaks is multimodal. In discrete series, mode is identified by direct inspection or through grouping tables and analysis tables when frequencies are irregular.

For grouped continuous frequency series with equal class intervals, the modal class corresponds to the class interval with the highest frequency \(f_1\). The modal value is estimated using the formula:

$$M_o = L + \left( \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \right) \times h$$
Where \(L\) is the lower limit of the modal class, \(f_1\) is the frequency of the modal class, $f_0$ is the frequency of the pre-modal class, \(f_2\) is the frequency of the post-modal class, and $h$ is the class width. An alternative formulation using absolute differences is \(M_o = L + \left( \frac{\Delta_1}{\Delta_1 + \Delta_2} \right) \times h\), where \(\Delta_1 = f_1 – f_0$ and $\Delta_2 = f_1 – f_2\). Graphically, mode is determined using a histogram by drawing lines from the top corners of the highest rectangle to the adjacent upper corners of neighboring rectangles and projecting the intersection point to the horizontal axis.

The spatial relationship between mean, median, and mode serves as a primary indicator of frequency curve symmetry and skewness. In a perfectly symmetric, unimodal normal distribution, all three measures coincide at a single point:

$$\text{Mean} = \text{Median} = \text{Mode}$$
In a positively skewed distribution (right-skewed), extreme high values pull the arithmetic mean to the right tail, resulting in the structural order:

$$\text{Mean} > \text{Median} > \text{Mode}$$
In a negatively skewed distribution (left-skewed), extreme low values drag the arithmetic mean to the left tail, producing the relation:

$$\text{Mean} < \text{Median} < \text{Mode}$$
For unimodal distributions that exhibit moderate asymmetry, Karl Pearson established the empirical relationship:

$$\text{Mode} = 3 \times \text{Median} – 2 \times \text{Mean}$$
Rearranging this formula yields expressions for median and mean:

$$\text{Median} = \frac{\text{Mode} + 2 \times \text{Mean}}{3}, \quad \text{Mean} = \frac{3 \times \text{Median} – \text{Mode}}{2}$$
This empirical rule underpins Karl Pearson’s second coefficient of skewness (\(S_k\)):

$$S_k = \frac{\text{Mean} – \text{Mode}}{\sigma} = \frac{3(\text{Mean} – \text{Median})}{\sigma}$$
Where \(\sigma\) is the standard deviation.

Comparative Properties and Economic Suitability

Criterion / Feature Arithmetic Mean Median Mode
Type of Measure Mathematical Average Positional Average Positional Average
Sensitivity to Outliers High (Extreme values distort value) Robust (Insensitive to extreme values) Robust (Insensitive to extreme values)
Open-Ended Classes Cannot be computed without assumptions Can be readily computed Can be readily computed
Algebraic Flexibility High (Combined mean, regression, sampling) None None
Graphical Determination Cannot be located graphically Determined using Ogives Determined using Histogram
Sampling Stability Highly stable across samples Moderately stable Least stable
Primary Economic Applications Per capita income, national accounts Wealth distribution, poverty lines Consumer preferences, market demand
The choice of central tendency measure in economic analysis depends on data structure, measurement scales, and objectives. The arithmetic mean is required when calculating aggregated national income metrics, cost functions, or econometric estimations, as it utilizes all sample data and permits advanced algebraic manipulation. However, because the arithmetic mean is sensitive to extreme values, it presents a misleading depiction of central tendency in highly skewed distributions such as personal income or wealth distributions.

In microeconomic welfare studies, poverty measurement, and wage distributions, the median is preferred because it is unaffected by extreme high-income outliers. The median operates effectively on ordinal data and open-ended class intervals (e.g., “below $10,000” or “above $500,000”) where upper or lower continuous boundaries are unspecified.

The mode is utilized in business economics, production planning, and market research to identify modal consumer demand, standard sizes in manufacturing, and high-frequency market trends. However, its utility is constrained by its vulnerability to sampling fluctuations, non-uniqueness in bimodal distributions, and complete lack of algebraic properties.

Skewness

Skewness is a statistical measure that describes the degree and direction of asymmetry in a frequency distribution around its central value. A distribution is symmetric when the observations are distributed equally on both sides of a central point. When the distribution is not symmetric, it is said to be skewed. Skewness therefore indicates whether the distribution has a longer or heavier tail toward the right or toward the left, and also indicates the extent of such asymmetry. In a perfectly symmetric distribution, the two sides of the distribution are mirror images of each other and the measure of skewness is zero. Skewness is a measure of the shape of a distribution, whereas measures such as mean, median and mode primarily describe its location.

A distribution may be positively skewed, negatively skewed, or approximately symmetric. In a positively skewed distribution, the tail extends farther toward the higher values. It is also called right-skewed distribution. In a negatively skewed distribution, the tail extends farther toward the lower values. It is also called left-skewed distribution. The terms positive and negative refer to the direction of asymmetry, not to whether the observations themselves are positive or negative.

In a symmetric distribution, the left and right portions of the distribution are approximately equal in shape. If the distribution is unimodal and perfectly symmetric, then:

$$
\text{Mean}=\text{Median}=\text{Mode}
$$

and the skewness is:

$$
Sk=0
$$

A normal distribution is the most important example of a perfectly symmetric distribution. In a symmetric distribution, the mean, median and mode coincide at the centre of the distribution. For a distribution that is only approximately symmetric, these three measures may be very close to one another even if they are not exactly equal.

In a positively skewed distribution, a relatively small number of observations with very high values pull the arithmetic mean toward the right-hand tail. The usual relationship is:

$$
\text{Mode}<\text{Median}<\text{Mean}
$$

Thus, the mean is generally greater than the median, and the median is generally greater than the mode. The right tail is longer than the left tail. Examples can include distributions of income, wealth, property prices, or waiting times where a small number of observations have exceptionally high values. The existence of a positive skew does not require every observation to be positive; positive skewness concerns the shape and direction of the tail.

In a negatively skewed distribution, a relatively small number of observations with very low values pull the arithmetic mean toward the left-hand tail. The usual relationship is:

$$
\text{Mean}<\text{Median}<\text{Mode}
$$

The left tail is longer than the right tail. Thus, the mean is generally smaller than the median, while the mode is generally larger than the median.

The basic distinction can be represented as follows:

Distribution Direction of longer tail Usual relationship
Symmetric Neither Mean = Median = Mode
Positively skewed Right Mode < Median < Mean
Negatively skewed Left Mean < Median < Mode

The mean, median and mode relationship is particularly important in identifying the direction of skewness. The mean is more sensitive to extreme observations than the median and mode. Therefore, extreme observations tend to pull the mean in the direction of the tail. This is the principal reason for the usual ordering of mean, median and mode in skewed distributions.

Skewness differs from dispersion. Dispersion measures the extent to which observations are spread around a central value, whereas skewness measures asymmetry. Standard deviation, variance and range are measures of dispersion. Skewness does not directly measure how widely observations are scattered; it measures whether the distribution is balanced or unbalanced around its centre.

Skewness also differs from kurtosis. Skewness concerns asymmetry, whereas kurtosis concerns the shape of the distribution in relation to the concentration of observations and the tails. A distribution can have zero skewness but still differ substantially from the normal distribution in its kurtosis.

Measures of Skewness:

Skewness can be measured through several statistical coefficients. The important measures include Karl Pearson’s coefficient of skewness, Bowley’s coefficient of skewness, Kelly’s coefficient of skewness, and moment coefficient of skewness. These measures are dimensionless coefficients and allow distributions to be compared even when they are measured in different units.

Karl Pearson’s Coefficient:

Karl Pearson’s coefficient of skewness is one of the most commonly used measures of skewness. When the mode is known, it is calculated as:

$$
Sk=\frac{\bar X-\text{Mode}}{\sigma}
$$

where:

  • \(\bar X\) = arithmetic mean

  • \(\text{Mode}\) = mode

  • \(\sigma\) = standard deviation

The numerator \(\bar X-\text{Mode}\) measures the difference between the mean and mode, while division by standard deviation converts the measure into a unit-free coefficient.

If:

$$
\bar X>\text{Mode}
$$

then:

$$
Sk>0
$$

and the distribution is positively skewed.

If:

$$
\bar X<\text{Mode}
$$

then:

$$
Sk<0
$$

and the distribution is negatively skewed.

If:

$$
\bar X=\text{Mode}
$$

then:

$$
Sk=0
$$

provided the distribution satisfies the relevant conditions.

In many practical distributions, the mode may not be clearly defined or may not be reliable. In such cases, Pearson’s second coefficient is used. It is:

$$
Sk=\frac{3(\bar X-\text{Median})}{\sigma}
$$

This formula is derived from the empirical relationship among mean, median and mode:

$$
\text{Mode}\approx3\text{Median}-2\bar X
$$

Rearranging:

$$
\bar X-\text{Mode}

\bar X-(3\text{Median}-2\bar X)
$$

Therefore:

$$
\bar X-\text{Mode}

3(\bar X-\text{Median})
$$

and hence:

$$
Sk=\frac{3(\bar X-\text{Median})}{\sigma}
$$

The Pearson coefficient is particularly useful for moderately skewed distributions. Its value is generally interpreted as positive, negative or zero according to the direction of asymmetry. The coefficient is not measured in the original units of the variable.

Bowley’s Coefficient:

Bowley’s coefficient of skewness is based on quartiles and the median. It is particularly useful when the distribution contains extreme observations because quartiles are less affected by extreme values than the arithmetic mean.

Bowley’s coefficient is:

$$
Sk_B=
\frac{Q_3+Q_1-2M}{Q_3-Q_1}
$$

where:

  • \(Q_1\) = first quartile

  • \(Q_2=M\) = median

  • \(Q_3\) = third quartile

The numerator:

$$
Q_3+Q_1-2M
$$

measures the asymmetry between the two halves of the distribution around the median.

The denominator:

$$
Q_3-Q_1
$$

is the interquartile range.

Therefore:

$$
Sk_B=
\frac{(Q_3-M)-(M-Q_1)}
{(Q_3-M)+(M-Q_1)}
$$

This form makes the interpretation particularly clear. If the distance from the median to \(Q_3\) is greater than the distance from \(Q_1\) to the median, the distribution is positively skewed. If the distance from \(Q_1\) to the median is greater, the distribution is negatively skewed.

If:

$$
Q_3-M>M-Q_1
$$

then:

$$
Sk_B>0
$$

and the distribution is positively skewed.

If:

$$
Q_3-M<M-Q_1
$$

then:

$$
Sk_B<0
$$

and the distribution is negatively skewed.

If:

$$
Q_3-M=M-Q_1
$$

then:

$$
Sk_B=0
$$

and the distribution is symmetric with respect to the quartiles and median.

An important property of Bowley’s coefficient is that it is less affected by extreme values because it uses \(Q_1\), median and \(Q_3\) rather than the mean and standard deviation. It is therefore useful for distributions containing open-ended class intervals, where calculation of the mean or standard deviation may be difficult or inappropriate.

Bowley’s coefficient is particularly useful for ordinal data or distributions in which the median and quartiles are more meaningful than the arithmetic mean. It does not utilise all observations directly.

The coefficient can also be expressed using the semi-interquartile range:

$$
Q.D.=\frac{Q_3-Q_1}{2}
$$

but Bowley’s coefficient itself is normally written as:

$$
Sk_B=
\frac{Q_3+Q_1-2Q_2}{Q_3-Q_1}
$$

since \(Q_2=M\).

Kelly’s Coefficient:

Kelly’s coefficient of skewness uses deciles or percentiles and therefore incorporates more information from the distribution than Bowley’s quartile-based measure.

Using deciles, Kelly’s coefficient is:

$$
Sk_K=
\frac{D_9+D_1-2D_5}{D_9-D_1}
$$

where:

  • \(D_1\) = first decile

  • \(D_5\) = fifth decile, which is the median

  • \(D_9\) = ninth decile

Since:

$$
D_5=M
$$

the formula can also be written as:

$$
Sk_K=
\frac{D_9+D_1-2M}{D_9-D_1}
$$

Using percentiles, it can be expressed as:

$$
Sk_K=
\frac{P_{90}+P_{10}-2P_{50}}
{P_{90}-P_{10}}
$$

where:

$$
P_{50}=M
$$

Kelly’s coefficient is useful when information about deciles or percentiles is available. Compared with Bowley’s coefficient, it uses a wider range of the distribution.

The principal difference is that Bowley’s measure concentrates on the central 50 percent of observations through \(Q_1\), \(M\), and \(Q_3\), whereas Kelly’s measure uses the central 80 percent through \(D_1\), \(D_5\), and \(D_9\).

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