Statistics: Mean, Median, Ogive, Mode

When dealing with large volumes of raw quantitative data (such as test marks of 1,000 students or daily wages of 500 factory workers), analyzing individual numbers is practically impossible. Human cognition requires a single central value around which the vast majority of observations cluster. This statistical representative is known as a Measure of Central Tendency (কেন্দ্রীয় প্রবণতার পরিমাপ).

In statistical tables, class intervals may be given in two formats:

• Exclusive Format (Continuous): e.g., $10-20, 20-30, 30-40$. The upper limit of one class equals the lower limit of the next. In this format, class limits and class boundaries are identical.
• Inclusive Format (Discontinuous): e.g., $10-19, 20-29, 30-39$. There is a gap of $1$ unit between classes. Before calculating the median or mode, you MUST convert inclusive limits into continuous class boundaries by subtracting $0.5$ from the lower limit and adding $0.5$ to the upper limit: $9.5-19.5, 19.5-29.5, 29.5-39.5$.

For continuous grouped data, first determine the class mark (mid-value) $x_i$ for each class interval:

Multiply each class mark $x_i$ by its corresponding frequency $f_i$, sum the products, and divide by the total frequency $N = \sum f_i$:

When values of $x_i$ and $f_i$ are large numbers, multiplying them directly becomes tedious and prone to arithmetic calculation errors. To simplify:

1. Choose a convenient middle class mark as the Assumed Mean, denoted by $a$.
2. Calculate the deviation $d_i$ for each class: $$d_i = x_i - a$$
3. Compute the product $f_i d_i$ and sum them: $\sum f_i d_i$.
4. The true mean is given by:
   $$\mathbf{ar{x} = a + rac{\sum f_i d_i}{\sum f_i} = a + rac{\sum f_i d_i}{N}}$$

When all class intervals have equal class length $h$, the deviations $d_i = x_i - a$ are all exact multiples of $h$. We scale down the deviations by dividing by $h$:

This reduces the values of $u_i$ to very small single-digit integers (e.g. $-3, -2, -1, 0, 1, 2, 3$). The step-deviation mean formula is:

Mathematical Proof: Since $u_i = rac{x_i - a}{h} \implies x_i = a + h u_i$. Multiplying by $f_i$ and summing: $\sum f_i x_i = \sum f_i (a + h u_i) = a\sum f_i + h\sum f_i u_i$. Dividing both sides by $N = \sum f_i$ yields $ar{x} = a + h rac{\sum f_i u_i}{N}$.

Arrange all $N$ observations in strictly ascending order:

• If $N$ is ODD: $ ext{Median} = \left(rac{N + 1}{2} ight) ext{-th observation}$.
• If $N$ is EVEN: $ ext{Median} = rac{1}{2} \left[ \left(rac{N}{2} ight) ext{-th observation} + \left(rac{N}{2} + 1 ight) ext{-th observation} ight]$.

For continuous grouped data, the exact median is calculated using the standard WBBSE interpolation formula:

1. Form the Cumulative Frequency Table: Compute the cumulative frequencies ($cf$) in ascending order.
2. Find Half the Total Frequency: Calculate $rac{N}{2}$, where $N = \sum f_i$.
3. Identify the Median Class: Locate the class interval whose cumulative frequency is just greater than or equal to $rac{N}{2}$. This interval is the Median Class.
4. Extract Formula Parameters:
   • $L$ = Lower class boundary of the median class.
   • $N$ = Total frequency ($\sum f_i$).
   • $cf$ = Cumulative frequency of the class preceding the median class.
   • $f$ = Simple frequency of the median class itself.
   • $h$ = Class width (length) of the median class ($h = ext{Upper Boundary} - ext{Lower Boundary}$).
5. Substitute the parameters into the formula and evaluate.

An Ogive (ওজাইভ) is a smooth cumulative frequency curve plotted on graph paper:

There are two standard graphical methods to obtain the Median:

• Method 1 (From a Single Less-than Ogive): Locate the point corresponding to $y = rac{N}{2}$ on the vertical y-axis. Draw a horizontal line parallel to the x-axis to intersect the less-than ogive at point $P$. From $P$, drop a vertical perpendicular line to the horizontal x-axis. The value of $x$ at the foot of this perpendicular is the Median.
• Method 2 (From the Intersection of Both Ogives): Plot both the less-than ogive and more-than ogive on the same graph sheet with identical axes. The two curves will intersect at a unique point $P$. Drop a perpendicular from $P$ to the x-axis. The point of intersection on the x-axis gives the Median directly!

The Mode (সংখ্যাগুরুমান) of a statistical dataset is the value that occurs with the greatest frequency. For continuous grouped frequency distributions:

1. Inspect the frequency column and identify the class interval having the maximum frequency. This interval is the Modal Class.
2. The mode of the distribution lies inside this modal class and is computed via the standard WBBSE modal formula:

• $L$ = Lower class boundary of the modal class.
• $f_1$ = Frequency of the modal class itself (the peak frequency).
• $f_0$ = Frequency of the class preceding the modal class.
• $f_2$ = Frequency of the class succeeding the modal class.
• $h$ = Class width (length) of the modal class ($h = ext{Upper Boundary} - ext{Lower Boundary}$).

For a moderately asymmetrical (unimodal, skewed) continuous distribution, the eminent statistician Karl Pearson discovered the fundamental empirical relationship connecting Mean, Median, and Mode:

If any two of the three measures are known, the third can be estimated immediately using this relationship.

In high-yield Madhyamik examination questions, one or two frequencies (often designated as $f_1, f_2$ or $x, y$) are missing from the table, but the Mean, Median, or Mode and the total frequency $N$ are given. The systematic solution workflow is:

1. Equation 1 (Total Frequency): Sum all frequencies including the unknown variables and set them equal to the given total frequency $N$: $$\sum f_i = ext{Sum of knowns} + x + y = N \implies x + y = N - ext{Sum of knowns}$$
2. Equation 2 (Parameter Formula):
   • If Median is given: Identify the median class directly because the given median value must lie inside its boundaries. Set up the median formula equation in terms of $x$.
   • If Mean is given: Use the Direct or Assumed Mean formula to set up an algebraic equation involving $x$ and $y$.
3. Solve the two linear simultaneous equations simultaneously to determine the exact integer values of $x$ and $y$.