Variance of ungrouped data

Ungrouped data refers to the data that one initially gathers from a study or an experiment. The said data is always raw. In other words, it is not sorted into any category, grouped, or otherwise classified. So, the ungrouped data is just a list of numbers. On the other hand, group data is the one that has been bundled into categories. This type of data is shown using frequency tables and histograms. Ungrouped data is also called raw data. It is always obtained from direct observation. We need the attributes of observation and frequency to show the tabular representation of ungrouped data.

A brief on Grouped Data

Grouped data refers to the type of data classified into groups after collection. The same raw data is categorised into various groups. After that, you need to create a table. The primary purpose of this particular table is to display the data points occurring in each group.

For example, after having done the test, the results are the data in this particular scenario. Moreover, there are many ways to group the data. For instance, you can record the number of students who scored above every 50 marks.

You can use the grades alternatively. As mentioned above, you can use a frequency table and histogram to calculate this type of data.

Advantages of Grouping Data

Grouping of data has multiple advantages. The main ones are given below:

1. Grouped data helps to improve the efficiency of estimations.
2. It allows better balancing of the statistical power of tests of the differences between strata. That is done by analysing an equal number from the strata.
3. Grouped data focuses on only the significant subpopulations and ignores the irrelevant ones.

A brief on Ungrouped Data

Ungrouped or raw data is not placed in any group of categories after the collection. So, the data is categorised either in numbers or characteristics. In other words, the data that is not put into either of the categories is ungrouped data. For example, you begin conducting a census. So, you want to analyse the number of women over 45 years in a particular area. 

You must know the total number of people who reside in that same area. So, the number of residents residing in that area account for raw information or ungrouped data. That is because nothing has been categorised. Now, we can conclude that ungrouped data is used to show information on an individual member of a population or a sample. There is a unique formula for calculating the variance of ungrouped data.

Advantages of Ungrouped Data

Following are the advantages of using ungrouped data:

1. Most people can easily interpret ungrouped data.
2. It is easy to calculate median, mean, and mode when the sample size is smaller.
3. No technical expertise is required to analyse ungrouped data.

Variance Formula

Variance is also represented by the symbol σ2. It refers to the squared variation of the values (xi) of a random variable (x) from the mean (μ). The variance formula helps us measure the spread from the mean of the random variable. This formula is different for a sample and a population and for both grouped and ungrouped data.

Steps for calculating the Variance of Ungrouped Data and Grouped Data

Given below are the simple steps for calculating the variance of ungrouped data and grouped data:

1. First, find the mean of the given data set. After adding all data values, divide them by the sample size (n).
2. Then, find the squared difference from the mean of each of the data values. Then subtract the mean from the data values and square the result.
3. Later, find the sum of all the squared differences.
4. Lastly, calculate the variance.

The variance of Ungrouped Data

The variance of ungrouped data is calculated as follows:

1. Population variance of ungrouped data – σ2=i=1n(xi −μ)2N

Here,

σ2 refers to the variance,

xi refers to the ith observation of the given data,

μ refers to the population mean,

N refers to the total number of observations or population size.

Sample variance of ungrouped data – S2=i=1n(xi − x )2n-1

Here,

S2 refers to the sample variance,

xi refers to the ith observation of the given data,

x refers to the sample mean,

n refers to the sample size or number of data values in the sample.

Variance of Grouped Data

After calculating the variance of ungrouped data, let us see how to calculate the variance of grouped data.

1.   Population variance of grouped data – σ2 = ∑ f (m − x̅)2 / n
2.   Sample variance of grouped data – s2 = ∑ f (m − x̅)2 / n – 1

Here,

f refers to the frequency of the class,

m refers to the midpoint of the class

x̅ refers to the sample mean,

n refers to the sample size or number of data values in the sample.

Conclusion

Both grouped and ungrouped data have striking differences. The same goes for their respective variance formula. The above study material notes on the variance of ungrouped data have taught us its advantages, calculations, and formulae. Ungrouped data or raw data has multiple advantages, the most important one being the flexibility in interpreting it.

 Almost anyone can collect ungrouped data because it is never grouped or categorised. So, you can collect it in raw form and later use it in your calculations as per convenience. The variance of ungrouped data is also very easy to calculate. You just need to find the mean initially and then add the data values, including the sum of the squared differences.