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Introduction
We will learn how to calculate the mean, variance, and standard deviation of grouped and ungrouped data. The variance calculator can be used to calculate the variance, standard deviation, and mean from a set of data with the sample size n to understand how far the observations deviate from the mean. The deviations can be either positive or negative. We must square the values to prevent positive and negative values from cancelling each other when we add up all the deviations.
Types of Data
Data is defined as the collection of facts or information from which conclusions can be drawn. The collected information can be numbers, words, measurements, and more.
There are several ways to classify data. One way to classify data is in terms of grouped data and ungrouped data.
Ungrouped Data
Ungrouped data is data that is not segregated into categories. It is also known as raw as it is not aggregated.
Example 1: Height of students
(177, 144, 151, 175, 166, 142, 163, 170, 161, 172, 162, 158, 190, 151, 187, 151, 160, 177, 148, 164)
Grouped Data
Data is referred to as grouped data when the raw data is organised into categories.
Example 2: Height of students in a table
Interval (in cms) | No. of Students (Frequency) | Cumulative Frequency |
141 – 150 | 4 | 4 |
151 – 160 | 4 | 8 |
161 – 170 | 6 | 14 |
171 – 180 | 4 | 18 |
181 – 190 | 2 | 20 |
Total No. of Students | – | 20 |
Mean
What is mean?
For a given set of data, the mean or arithmetic mean is the average of all the values in the data set. The mean of the data set reflects the characteristics of all the data. It also reflects the data’s distribution relative to the central part of the distribution.
An average should always be greater than the lowest value in the data set and lower than the largest value in the data set. Ideally, the mean should be in the centre of the data set.
Arithmetic mean can be calculated using the formula:
Mean, x = Sum of all ObservationsNumber of Observations
How to calculate mean for ungrouped data?
To calculate the mean for ungrouped data, we will use the data from Example 1: Height of students. Thus, the mean is calculated as:
x=Sum of all ObservationsNumber of Observations= (x)f
=(177, 144, 151, 175, 166, 142, 163, 170, 161, 172, 162, 158, 190, 151, 187, 151, 160, 177, 148, 164)20=326020 =163.45 – (i)
How to calculate mean for grouped data?
To calculate the mean for grouped data, we will use the data from Example 2: Height of students given in a tabular form below and calculate the midpoints of the intervals. The sum of observations is calculated by multiplying the frequency with the midpoint of the interval.
Height | MidPoint (x) | No. of Students (f) | f * x |
141 – 150 | 145 | 4 | 580 |
151 – 160 | 155 | 4 | 620 |
161 – 170 | 165 | 6 | 990 |
171 – 180 | 175 | 4 | 700 |
181 – 190 | 185 | 2 | 370 |
Total | 825 | 20 | 3260 |
Therefore, by using the formula Mean, (x) = Sum of all Observations Number of Observations .
We get,
x= fxf=326020=163
Variance
What is variance?
The variance is an estimate of the degree of variability. It is determined by averaging the square of deviations from the mean. The degree of dispersion in the data is indicated by the variance calculator. The wider the data spread, the higher the variance from the mean.
How to calculate variance for ungrouped data?
The variance of a set of values, denoted by 2, is defined as
2= x2n -(x)2
Where x is the mean, n is the number of data values, and x2 stands for the sum of all data values.
Using the variance calculator to calculate variance (2), we refer to the ungrouped data from Example 1: Height of students.
Using (i), we get
x =163.45
Thus, 2= x2n -(x)2 =173.75 – (ii)
Where,
x2n = (177, 144, 151, 175, 166, 142, 163, 170, 161, 172, 162, 158, 190, 151, 187, 151, 160, 177, 148, 164)220
and n is the total number of students.
How to calculate variance for grouped data?
The variance for grouped data, which we denote by 2, is defined as
2= fx2n – (x)2
Where x = fxn
Here x is the mean, n is the number of data values, stands for the sum of, f is the frequency of the data value, and x stands for each data value. Note: f = n.
By using the variance calculator to calculate variance (2), we refer to the data from Example 2: Height of students given in a tabular form and calculate the midpoints of the intervals.
Hence, we get the following table:
Height | Mid Point (x) | x2 | No. of Students (f) | f * x | f * x2 |
141 – 150 | 145 | 21025 | 4 | 580 | 84100 |
151 – 160 | 155 | 24025 | 4 | 620 | 96100 |
161 – 170 | 165 | 27225 | 6 | 990 | 163350 |
171 – 180 | 175 | 30625 | 4 | 700 | 122500 |
181 – 190 | 185 | 34225 | 2 | 370 | 68450 |
Total | 825 | 137125 | 20 | 3260 | 534500 |
Here, mean, x = fxn=326020=163
Therefore, variance = 2= fx2n–(x) 2
=53450020–1632=156 – (iii)
Standard Deviation
What is standard deviation?
The standard deviation is calculated as the square root of variance. Standard deviation helps us understand the degree of deviation of value from the mean value of the data set. A small standard deviation value implies that the data is near the mean, whereas a large standard deviation value indicates a significant distance between the mean and the data values.
How to calculate the standard deviation for ungrouped data?
The standard deviation for ungrouped data is denoted by = x2n – (x)2
Where x = mean, n = number of data values, and x2 = sum of all data values.
In other words, standard deviation, =variance
Thus, from (ii), we get standard deviation =173.75=13.18
How to calculate the standard deviation for grouped data?
The standard deviation for grouped data, which we denote by , is defined as
= fx2n-(x)2
Where x = fxn
Here, x is the mean, n is the number of data values, stands for the sum of, f is the frequency of the data value, and x stands for each data value. Note: f = n
In other words, standard deviation, =variance for grouped data.
Thus, from (iii) we get, standard deviation, =156= 12.49
Conclusion
As mentioned above, ungrouped data is the raw data collected in its actual form. The distribution of this raw data into different classes is known as grouped data.
It is important to evaluate both variance and standard deviation to understand how much the data deviates from the mean. The concept used for calculating the mean, variance, and standard deviation of both grouped data and ungrouped data is the same. However, the formulas are different because of the difference in the representation of the data in the two data categories.
