CBSE Class 10 Mathematics Statistics Notes Set 06

Median

Median is the middle value of the distribution. It is the value of the variable such that the number of observations above it is equal to the number of observations below it.

(a) Median of raw data

Step 1: Arrange the data in ascending order.
Step 2: Count the number of observations (let there be n observations).
1. If n is odd, then median = value of the \( \left( \dfrac{n + 1}{2} \right)^{\text{th}} \) observation.
2. If n is even, then median is the arithmetic mean of the \( \left( \dfrac{n}{2} \right)^{\text{th}} \) observation and the \( \left( \dfrac{n}{2} + 1 \right)^{\text{th}} \) observation.

(b) Median of discrete frequency distribution

In case of a discrete frequency distribution, we calculate the median by using the following steps.
Step 1: Find the cumulative frequencies (c.f.).
Step 2: Find \( \dfrac{N}{2} \), where N = Σfᵢ.
Step 3: See the cumulative frequency (c.f.) just greater than \( \dfrac{N}{2} \) and determine the corresponding value of the variable. This value is the required median.

(c) Median of class interval data (grouped)

In a grouped data, we may not find the middle observation by looking at the cumulative frequencies, since the middle observation will be some value in a class interval, so it is necessary to find the value inside a class interval that divides the whole distribution into two halves. For this, we find the cumulative frequencies of all the classes and \( \dfrac{N}{2} \), where N = number of observations. Now locate the class whose cumulative frequency is greater than (and nearest to) \( \dfrac{N}{2} \). This class is called the median class. After finding the median class, use the following formula for calculating the median.

\[ \text{Median} = \ell + \dfrac{\dfrac{N}{2} - C}{f} \times h \] where ℓ = lower limit of median class, N = total number of observations, C = cumulative frequency of the class preceding the median class, h = size of the median class and f = frequency of the median class.

(d) Merits of median

1. It is rigidly defined, easily understood and calculated.
2. It is not at all affected by extreme values.
3. It can be located graphically, even if the class intervals are unequal.
4. It can be determined even by inspection in some cases.

(e) Demerits of median

1. In case of an even number of observations the median cannot be determined exactly.
2. It is not based on all the observations.
3. It is not subject to algebraic treatment.
4. It is much affected by fluctuations of sampling.

(f) Uses of median

1. Median is the only average to be used while dealing with qualitative data which cannot be measured quantitatively but can be arranged in ascending or descending order of magnitude.
2. It is used for determining the typical value in problems concerning wages, distribution of wealth etc.

Solved Examples

Example 9. Following are the lives in hours of 15 pieces of the components of an aircraft engine. Find the median: 715, 724, 725, 710, 729, 745, 649, 699, 696, 712, 734, 728, 716, 705, 719.
Answer:
Arranging the data in ascending order: 649, 696, 699, 705, 710, 712, 715, 716, 719, 724, 725, 728, 729, 734, 745. Here N = 15.
Median = \( \left( \dfrac{N + 1}{2} \right)^{\text{th}} \) observation = \( \left( \dfrac{15 + 1}{2} \right)^{\text{th}} \) observation = 8th observation = 716.

Check Your Progress

1. Find the median of 9, 5, 7, 11, 13, 3.
2. The shirt sizes worn by a group of 198 persons are as follows. Find the median shirt size.

3. The number of words in each of the first eighty sentences of a book were counted. The results are recorded in groups as follows. Estimate the median length of sentence correct to one decimal place.

4. The weights of 80 parcels were each measured to the nearest tenth of a kilogram, and the results recorded in groups as follows. Calculate the median weight.

Answers
1. 8
2. 40
3. 16.3
4. 33.4

Mode

Mode or modal value of the distribution is that value of the variable for which the frequency is maximum.

(a) Mode of ungrouped data (by inspection only)

Arrange the data in an array and then count the frequencies of each variate. The variate having maximum frequency is the mode.

(b) Mode of continuous frequency distribution

In case of a grouped frequency distribution, it is not possible to determine the mode by only looking at the frequencies of observations. In such a situation, we can only locate a class having maximum frequency. This class is called the ‘modal class’. The mode is obtained by using the value inside the modal class, calculated by the formula \[ \text{Mode} = \ell + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h \] where ℓ = lower limit of the modal class, f₁ = frequency of the modal class (the largest frequency), f₀ = frequency of the class preceding the modal class, f₂ = frequency of the class succeeding the modal class and h = width of the modal class.

(c) Merits of mode

1. It can be easily understood and is easy to calculate.
2. It is not affected by extreme values and can be found by inspection in some cases.
3. It can be measured even if open-end classes are present and can be represented graphically.

(d) Demerits of mode

1. It is ill-defined. It is not always possible to find a clearly defined mode.
2. It is not based upon all the observations.
3. It is not capable of further mathematical treatment. It is often indeterminate.
4. It is affected to a greater extent by fluctuations of sampling.

(e) Uses of mode

Mode is the average to be used to find the ideal size, e.g., in business forecasting, in manufacture of ready-made garments, shoes etc.

(f) Empirical relationship between the three measures of central tendency

The empirical relationship between mean, median and mode is Mode = 3 Median − 2 Mean. This relationship can be used to find a central tendency when the other two central tendencies are given.

Solved Examples

Example 16. If the mean and the median of the data are 54 and 51 respectively, then find the mode of the data.
Answer:
Mode = 3 Median − 2 Mean = 3 × 51 − 2 × 54 = 153 − 108 = 45.

Check Your Progress

1. Find the mode of 0, 0, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 7, 8.
2. Write the empirical relation between mean, mode and median.
3. In an examination taken by 400 students, the scores were as shown in the following distribution table. Find the mode score.

4. Find the mode of the following distribution.

5. If the median and mode are 9.98 and 10.09 respectively, then find the mean.

Answers
1. 5
2. Mode = 3 Median − 2 Mean
3. 47.26
4. 36.76
5. 9.925