Variance Formulas for Ungrouped Data

Variance Formulas for Ungrouped Data is a kind of distribution in which information is delivered to each individual in its entirety. For example, a batsman’s scores in the last five matches are as follows: 45, 34, 2, 77, and 80 runs. Calculating the range and mean deviation from this data will help determine his general shape and performance. Here, we will look at methods for estimating the range and mean deviation for individual series distributions.

The Standard Deviation and the Mean Deviation: Variance Formulas 

In simple words, the range of a distribution may be defined as the difference between the distribution’s greatest and smallest values. The phrase “mean deviation” may be defined mathematically as “the ratio of the sum of all the absolute values of dispersion to the number of observations.”

In layman’s words, mean deviation depicts the dispersion of all data points in a series in reference to the measure of central tendency, which we shall refer to as the median or mean throughout this article. Additionally, the mean deviation from the mode may also be calculated.

Ungrouped Data Sets: Variance Formulas

We now know that the range is defined as the difference between the highest and lowest potential values. As a result, while working with ungrouped data, we arrange the series in ascending or descending chronological order. This helps us choose the values at the top and bottom of the distribution. Going forward, we will simply subtract the lowest value from the greatest value to get the minimal value.

For example, a student’s marks in a five-chapter statistics test are (out of a potential 20) 11, 14, 16, 13, and 18 on the scale. They must be placed in the following order: 18, 16, 14, 13, and 11 are the digits. The data range is 18-11=7.

As previously mentioned, the mean deviation is a statistical measure of data dispersion around a central tendency measure. The most widely used metric of central tendency is the median or mean of central tendency. Let’s begin by going through how to calculate the mean and median for each distribution series in the first place.

Mean and Median: Variance Formulas

To compute the median, we must first arrange the data in ascending or descending order, depending on the circumstances (generally ascending order). The number of observations, denoted by the letter n, is also tallied in this stage. The following calculation is broken into two sections depending on whether n is an even or an odd number:

• If n is an odd number, the median is the value of (n+1)/the 2nd item on the list
• If n is an even number, the median is computed as [value of (n+1)/2th item + value of (n/2 +1)th item]
• The mean is simply determined as the ratio of the total number of observations to the sum of observations in a set of observations
• The mean is defined as the sum of observations divided by the total number of observations

How to Determine the Mean Deviation?

After calculating the mean and median, we can go on to calculate the mean deviation. The value with respect to which the mean deviation is calculated is represented by the letter ‘a.’ The modulus (absolute value) of the difference between ‘a’ and the data member in the previous phase is used to calculate further deviations for the data members. The symbol |X – a| represents the deviation of a value X.

Finally, the sum of these variances for all data members is divided by the number of observations, denoted by the letter n, to get the final result.

[|X – a|]n is the mean deviation.

In this scenario, |X-a| = the sum of the deviations for all values starting with ‘a’.

Variance Formulas:  Formula for Sample Variance

Before we examine the variance formula, let us discuss what variance is. When a random variable (X) contains values (Xi), the variance (2) is the square of the difference in squares between those values (Xi). The variance formula enables us to determine the spread between a random variable mean and variance. A population’s variance formula is distinct from a sample’s variance formula. Take a look at the next section’s variance formula.

Conclusion:   Variance Formulas for Ungrouped Data

We have seen that the distance between the data’s centre and the data’s mean deviation may be calculated using this formula. The data set’s centre points are the mean, median, and mode. In other words, the average of the data’s absolute departures from the centre point is calculated using the mean deviation. For both grouped and ungrouped data, a mean deviation may be determined.

As contrast to standard deviation, mean deviation is a more user-friendly way to assess variability. The mean deviation is a useful tool for estimating the data’s average skewness.