The Variance Of Grouped Data

Statistical analysis is the method of carrying out the collection and summarization of data. It can be defined as the discipline of mathematics associated with the process of collection, analysis, interpretation, organization, and presentation of data in a particular and concise manner. The data sets are usually represented in a tabular format for easy understanding with numerical.

Various fields of mathematics are associated and involved with statistics, such as linear algebra, differential equations, measure-theoretic probability theory, and stochastic analysis.

Statistics has a high value in various math-oriented sectors such as probability, sociology, psychology, geology, and even weather forecasting.

Statistical Analysis

Statistics is a simple tool that can process and improve data quality by developing specific experimental designs and survey samples.

Statistical analysis can be divided into two types

1. Descriptive statistics
2. Inferential statistics

Descriptive statistics

In this field of statistical analysis, we describe the collection of data in a summarised method.

Inferential statistics

In this field of statistical analysis, we derive the inference required to explain the descriptive data.

Both these fields of statistics are used on a large scale, and both can be combined too.

Measures of Dispersion

An important concept of statistics deals with various disciplines such as finance, business, and accounting. They can also be called measures of central tendency.

The different types of measures of dispersion are listed under:

• Arithmetic mean
• Harmonic mean
• Geometric mean
• Median and mode
• Partition values
• Range and mean deviation
• Quartile deviation
• Standard deviation
• Variance

Standard Deviation

Standard deviation measures the amount of variation or dispersion of a set of collective data. If the standard deviation value is low, it indicates that the given values are close to the mean of the set. If the standard deviation value is high, the values are spread out over a wide range.

The Standard Deviation (σ)= √[∑d²/N

Where d = (x-x1)2

σ = population standard deviation

N = size of the population

D= deviation of an item relative to the mean

x= each value from the population

x1 = population mean

The standard deviation of grouped data

The formula for calculating the standard deviation in a grouped frequency distribution is

σ = [(Σi fi (yi – ȳ)2 ⁄ N] ½ 

Standard deviation can never be a negative value.

Variance

Variance is defined as a measure of dispersion that provides the extent of variability between the numbers in a collective data set. It is used to measure the distance apart from each number in the set from the mean. Analysts and traders utilize the term variance to determine volatility and market security.

Variance is of two types, namely, population variance and sample variance.

Population variance

If you wish to calculate the variance for a whole population, the concept of population variance can be used. To get an exact value, you need to collect data from every member of the population.

The population variance formula is given as:

σ2 = ∑ (X-μ)2 /N

σ2 = population variance

Χ = each value

μ = population mean

Ν = number of observations in the population

Sample variance

If you wish to calculate the variance for a particular sample, the concept of sample variance can be used. The sample variance can also be used for making estimates or inferences about the population variance.

The formula for sample variance is given as:

s2 = ∑ f(x-x1)2 /n-1

s2 = sample variance

x = each value

x1 = sample mean

n = number of values in the sample

The variance of grouped data

The formula for calculating the variance in a grouped frequency distribution is:

σ2 = [(Σi fi (xi –x1)2 ⁄ N] = [(Σi fi xi2 ⁄ n) – (Σi fi xi/N)2] (population variance)

Where, xi=midpoint of the ith class

Conclusion

Variance is an important statistical analysis to be considered before performing parametric tests. The variance of grouped data can be calculated using the suitable formula described above. Variance and standard deviation are used to visualize and understand the interpretation of the data being studied. The value of variance is always zero or positive and never negative. Zero value of variance can indicate identical values. The study material notes on the variance of grouped data can be very useful to understand these concepts of variance.