There are three main measures of central tendency for statistics, the mean, median, and mode. The mean is the average of all values, determined by adding them together and then dividing by the count. The median is that middle value when the data are ordered, or the average of the two middle values if there's an even number. That value which appears most often in the dataset is known as the mode. Each one gives a different insight into the data-mean as an average, median as a middle of data, and mode as the most frequent value.
Definition: The mean is generally known as the average. It gives a middle value to the data.
Here,
∑ represents the summation
X represents the observations
N represents the number of observations.
If the data is given in tabulated form, then the formula adopted to compute the mean is
Mean = ∑f x / ∑f
Where ∑f = N
Definition:The median is the middle value in an ordered dataset. If the total number of values is even then the median is taken as the average of the two middle values.
If the total number of observations, n is odd in number then the formula is as follows:
If the total no. of the observations, n is an even number, then the formula is as follows,
Consider the case in which the data are continuous and given in the form of frequency distribution, then the formula for median is as follows.
The class containing the median is known as the median class. The total frequency of all the observations ∑f.
The class containing (n / 2) is called the median class.
Here
l = lower limit belonging to the median class
c = cumulative frequency value of the class preceding the median class
f = frequency carried by the median class
h = class size
The mode is that value which occurs most often in a dataset. A distribution of data may have one mode, more than one mode, or no mode in the event that no value repeats.
Consider that the data are continuous and the value of mode can be calculated with the help of the following steps.
a] Identify the modal class that is the class having maximum frequency.
b] By using mode formula below
l = lesser limit of modal class
= frequency possessed by the modal class
= frequency possessed by the class prior the modal class
= frequency possessed by the class after the modal class
h = width of the class
Problems 1: Find the mean, median, mode and range for the following list of values:
13, 18, 13, 14, 13, 16, 14, 21, 13
Solution:
Given Value : 13, 18, 13, 14, 13, 16, 14, 13, 21
The mean is the usual average.
Mean = {13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13} / {9} = 15
The median is the middle value. Thus, you must rewrite the list in order from smallest to largest to find the middle value: 13, 13, 13, 13, 14, 14, 16, 18, 21
There are nine no.s in the list, so the middle one will be
{9 + 1} / {2} = {10} / {2} = 5
= 5th number
Thus, the median is 14.
The mode is the number occurring more frequently than any other; hence, 13 is the mode.
The largest number in the list is 21, and the smallest one is 13. Therefore, the range is 21 – 13 = 8.
Mean = 15
Median = 14
Mode = 13
Range = 8
Problem 2: The value of the mean of five numbers is observed to be 18.Therefore, if one number is not included, the mean is 16. Find the number that is excluded.
Solution:
From the question,
5 observations that mean n = 5.
The value of the mean = 18
x̄ = 18
x̄ = ∑ x / n
∑ x = 5 * 18 = 90
The sum of the five observations = 90.
Assume the excluded number to be “a”
Sum of four observations = 90 – a
Four observations of Mean = (90 – a) / 4
16 = (90 – a) / 4
90 – a = 64
a = 26
⇒ The excluded number is 26.