Economics: Class 11 (Part 2) — Chapter 5: Measures of Central Tendency

There are three methods to calculate the mean for ungrouped data:

• Direct Method:
  $$\bar{X} = \frac{\sum X}{N}$$ Where $\sum X$ is the sum of all observations, and $N$ is the total number of observations.
• Assumed Mean Method: Used when the number of observations is large to simplify calculations.
  $$\bar{X} = A + \frac{\sum d}{N}$$ Where $A$ is the assumed mean, and $d = X - A$ represents the deviations of observations from the assumed mean.
• Step Deviation Method: Further simplifies calculations by dividing deviations by a common factor $c$.
  $$\bar{X} = A + \frac{\sum d'}{N} \times c$$ Where $d' = \frac{X - A}{c}$ and $c$ is the common factor.

Discrete Series

• Direct Method:
  $$\bar{X} = \frac{\sum fX}{\sum f}$$ Where $f$ is the frequency of each observation $X$.
• Assumed Mean Method:
  $$\bar{X} = A + \frac{\sum fd}{\sum f}$$ Where $d = X - A$.
• Step Deviation Method:
  $$\bar{X} = A + \left(\frac{\sum fd'}{\sum f}\right) \times c$$ Where $d' = \frac{X - A}{c}$.

Continuous Series

In a continuous series, class intervals are given. The mid-points ($m$) of these class intervals are used as the representative values for calculation.

• Direct Method:
  $$\bar{X} = \frac{\sum fm}{\sum f}$$
• Step Deviation Method:
  $$\bar{X} = A + \left(\frac{\sum fd'}{\sum f}\right) \times c$$ Where $d' = \frac{m - A}{c}$ and $m$ is the mid-point of each class.

1. The algebraic sum of deviations of items from their arithmetic mean is always zero: $\sum (X - \bar{X}) = 0$.
2. The arithmetic mean is highly affected by extreme values (outliers).

When different items hold varying levels of importance, weights ($w$) are assigned to them. The weighted arithmetic mean is calculated as:

• Individual Series: Arrange the data in ascending or descending order.
  • If $N$ is odd, the median is the value of the $\left(\frac{N+1}{2}\right)^{\text{th}}$ item.
  • If $N$ is even, the median is the arithmetic mean of the two middle values: $\frac{\text{Value of } (N/2)^{\text{th}} \text{ item} + \text{Value of } (N/2 + 1)^{\text{th}} \text{ item}}{2}$.
• Discrete Series: Find the cumulative frequencies ($c.f.$). The median is the value of the variable corresponding to the cumulative frequency that contains the $\left(\frac{N+1}{2}\right)^{\text{th}}$ item.
• Continuous Series: Locate the median class where the cumulative frequency contains the $\left(\frac{N}{2}\right)^{\text{th}}$ item. The median is calculated using:
  $$\text{Median} = L + \frac{\frac{N}{2} - c.f.}{f} \times h$$ Where:

  • $L$ = lower limit of the median class
  • $c.f.$ = cumulative frequency of the class preceding the median class
  • $f$ = frequency of the median class
  • $h$ = magnitude/width of the median class interval

These partition values divide the distribution into more than two parts:

• Quartiles divide the data into four equal parts ($Q_1$, $Q_2$, $Q_3$).
  • First Quartile ($Q_1$): Size of $\frac{N+1}{4}^{\text{th}}$ item.
  • Third Quartile ($Q_3$): Size of $\frac{3(N+1)}{4}^{\text{th}}$ item.
• Percentiles divide the distribution into 100 equal parts ($P_1, P_2, \dots, P_{99}$). $P_{50}$ is equal to the median.

• Discrete Series: Identified by inspection as the value with the highest frequency.
• Continuous Series: Locate the modal class (class with the highest frequency). The mode is calculated using:
  $$M_o = L + \frac{D_1}{D_1 + D_2} \times h$$ Where:

  • $L$ = lower limit of the modal class
  • $D_1 = f_1 - f_0$ (difference between frequency of modal class and preceding class)
  • $D_2 = f_1 - f_2$ (difference between frequency of modal class and succeeding class)
  • $h$ = class interval width