CBSE Class 10 Mathematics Statistics Notes Set 10

Mean of Grouped Data

For a frequency distribution with class marks xᵢ and frequencies fᵢ (class mark = (lower limit + upper limit)/2):
Direct method: Mean = \( \frac{Σ f_i x_i}{Σ f_i} \)
Assumed mean method: Mean = a + \( \frac{Σ f_i d_i}{Σ f_i} \), where dᵢ = xᵢ − a and a is a convenient class mark near the middle.
Step deviation method: Mean = a + h × \( \frac{Σ f_i u_i}{Σ f_i} \), where uᵢ = (xᵢ − a)/h and h is the common class width.
Adding or subtracting a constant to every observation changes the mean by the same constant; multiplying or dividing every observation by a constant multiplies or divides the mean by it.

Median of Grouped Data

The median is the middle value. Steps: (1) form the cumulative frequency table, (2) find N/2, where N = Σf, (3) locate the median class, the first class whose cumulative frequency is at least N/2.
Median = l + \( \left(\frac{N/2 − cf}{f}\right) × h \)
where l is the lower limit of the median class, cf the cumulative frequency of the class just before it, f its frequency and h its width.
For ungrouped data, sort the values: if n is odd the median is the ((n + 1)/2)th value; if n is even it is the mean of the (n/2)th and (n/2 + 1)th values.

Mode of Grouped Data

The mode is the value that occurs most often. For grouped data, the modal class is the class with the highest frequency, and
Mode = l + \( \left(\frac{f_1 − f_0}{2f_1 − f_0 − f_2}\right) × h \)
where l is the lower limit of the modal class, f₁ its frequency, f₀ the frequency of the class before it, f₂ the frequency of the class after it and h the class width.

Relation Between the Three Averages

For a moderately skewed distribution, 3 Median = Mode + 2 Mean, that is Mode = 3 Median − 2 Mean.
For a symmetrical distribution, mean = median = mode.

Cumulative Frequency Curves (Ogives)

A 'less than' ogive plots the upper class limits against the 'less than' cumulative frequencies. A 'more than' ogive plots the lower class limits against the 'more than' cumulative frequencies.
The x-coordinate of the point where the two ogives cross is the median, and the y-coordinate there is N/2. The median can also be read by drawing a horizontal line from N/2 on the 'less than' ogive.

Worked Examples

Example 1. Find the mean of the data: class 0–10, 10–20, 20–30, 30–40 with frequencies 5, 10, 15, 10.
Solution: Class marks 5, 15, 25, 35. Σfx = 25 + 150 + 375 + 350 = 900, Σf = 40. Mean = 900/40 = 22.5.

Example 2. Find the median for the same data.
Solution: N = 40, N/2 = 20. Cumulative frequencies: 5, 15, 30, 40. Median class = 20–30, l = 20, cf = 15, f = 15, h = 10. Median = 20 + (5/15) × 10 = 23.33.

Example 3. Find the mode of the same data.
Solution: The modal class is 20–30 (frequency 15). l = 20, f₁ = 15, f₀ = 10, f₂ = 10, h = 10. Mode = 20 + (5/10) × 10 = 25.

Example 4. If the mode of a distribution is 8 and its mean is 7, find the median.
Solution: 3 Median = Mode + 2 Mean = 8 + 14 = 22, so Median = 22/3 ≈ 7.33.

Example 5. The mean of 25 observations is 36. If one observation 48 is replaced by 8, find the new mean.
Solution: Old total = 900. New total = 900 − 48 + 8 = 860. New mean = 860/25 = 34.4.