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Median, Quartiles and Mode | Mathematics (Maths) Class 11 - Commerce PDF Download

Median

The median of a distribution is the value of the middle variable when the variables are arranged in ascending or descending order.

Median (Md) is an average of position of the numbers.


1. Median for Simple Distribution

Firstly, arrange the terms in ascending or descending order and then find the number of terms n.

(a) If n is odd, then (n + 1 / 2)th term is the median.

(b) If n is even, then there are two middle terms namely (n / 2)th and (n / 2 + 1)th terms. Hence,

Median = Mean of (n / 2)th and (n / 2 + 1)th terms.


2. Median for Unclassified Frequency Distribution

(i) First find N / 2, where N = Σ fi.

(ii) Find the cumulative frequency of each value of the variable and take value of the variable which is equal to or just greater than N / 2

(iii) This value of the variable is the median.


3. Median for Classified Data (Median Class)

If in a continuous distribution, the total frequency be N, then the class whose cumulative frequency is either equal to N / 2 or is just greater than N / 2 is called median class.

For a continuous distribution, median

Md = l + ((N / 2 – C) / f) * h

where, l = lower limit of the median class

f = frequency of the median class

N = total frequency = Σ f

C = cumulative frequency of the class just before the median class

h = length of the median class


Quartiles

The median divides the distribution in two equal parts. The distribution can similarly be divided in more equal parts (four, five, six etc.). Quartiles for a continuous distribution is given by

Q1 = l + ((N / 4 – C) / f) * h

Where, N = total frequency

l = lower limit of the first quartile class

f = frequency of the first quartile class

C = the cumulative frequency corresponding to the class just before the first quartile class

h = the length of the first quartile class

Similarly, Q3 = l + ((3N / 4 – C) / f) * h

where symbols have the same meaning as above only taking third quartile in place of first quartile.


Mode

The mode (Mo) of a distribution is the value at the point about which the items tend to be most heavily concentrated. It is generally the value of the variable which appears to occur most frequently in the distribution.


1. Mode for a Raw Data

Mode from the following numbers of a variable 70, 80, 90, 96, 70, 96, 96, 90 is 96 as 96 occurs maximum number of times.

Median, Quartiles and Mode | Mathematics (Maths) Class 11 - Commerce


For Classified Distribution

The class having the maximum frequency is called the modal class and the middle point of the modal class is called the crude mode.

The class just before the modal class is called pre-modal class and the class after the modal class is called the post-modal class.


Mode for Classified Data (Continuous Distribution)

Mo = l + (f0 – f1 / 2 f0 – f1 – f2) x h

Where, 1 = lower limit of the modal class

f0 = frequency of the modal class

f1 = frequency of the pre-modal class

f2 = frequency of the post-modal class

h = length of the class interval


Relation between Mean, Median and Mode

(i) Mean — Mode = 3 (Mean — Median)

(ii) Mode = 3 Median — 2 Mean


Symmetrical and Skew distribution

A distribution is symmetric, if the same number of frequencies is found to be distributed at the same linear teance on either side of the mode. The frequency curve is bell shaped and A = Md = Mo

Median, Quartiles and Mode | Mathematics (Maths) Class 11 - Commerce

In anti-symmetric or skew distribution, the variation does not have symmetry.

(i) If the frequencies increases sharply at beginning and decreases slowly after modal value, then it is called positive skewness and A > Md > Mo.

Median, Quartiles and Mode | Mathematics (Maths) Class 11 - Commerce

(ii) If the frequencies increases slowly and decreases sharply after modal value, the skewness is said to be negative and A < Md < Mo.

Median, Quartiles and Mode | Mathematics (Maths) Class 11 - Commerce

The document Median, Quartiles and Mode | Mathematics (Maths) Class 11 - Commerce is a part of the Commerce Course Mathematics (Maths) Class 11.
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FAQs on Median, Quartiles and Mode - Mathematics (Maths) Class 11 - Commerce

1. What is the median and how is it calculated?
The median is a measure of central tendency that represents the middle value in a dataset. To calculate the median, you arrange the data in ascending order and find the value that falls exactly in the middle. If there is an odd number of data points, the median is the middle value. If there is an even number of data points, the median is the average of the two middle values.
2. How do quartiles divide a dataset?
Quartiles divide a dataset into four equal parts. The first quartile (Q1) is the value that separates the lowest 25% of the data from the remaining 75%. The second quartile (Q2) is the median, which divides the data into two equal halves. The third quartile (Q3) separates the lowest 75% of the data from the highest 25%.
3. What is the significance of quartiles in statistical analysis?
Quartiles play a crucial role in statistical analysis as they provide information about the spread and distribution of a dataset. They give insights into the range of values and help identify the presence of outliers. Quartiles are also used to calculate the interquartile range (IQR), which measures the dispersion of the middle 50% of the data.
4. How is the mode determined and what does it represent?
The mode is the value or values that occur most frequently in a dataset. To determine the mode, you analyze the frequency distribution of the data and identify the value(s) with the highest frequency. The mode represents the peak or peaks of the data, indicating the most common value(s) in the dataset.
5. Can a dataset have multiple modes?
Yes, a dataset can have multiple modes. If there are two or more values with the highest frequency in a dataset, it is considered multimodal. On the other hand, if no value is repeated or if all values have the same frequency, the dataset is considered to have no mode (no mode).
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