How To Find Mean Deviation For Ungrouped Data

Mean deviation is used to compute how some distance the values in a data set are from the center point. imply, median, and mode all form middle points of the statistics set. In different phrases, the mean deviation is used to calculate the common or the Mean Deviations of the facts from the important point. Mean Deviations can be calculated for each grouped and ungrouped statistic.

Mean Deviation is a simpler measurement of variability in comparison to conventional deviation. When we need to discover the Mean deviation from the statistics' center point, the mean deviation is used. In this article, we will take an in-intensity observation to Mean Deviation, its system, and examples in addition to the merits and demerits.

To understand the idea of mean deviation for ungrouped data, one should first take a glance on the concept of ‘Frequency Distribution’. A frequency distribution is a depiction of several observations within a given interval, either in the form of a graph or in a tabular format. The analyzed data and the objective of the analyst govern the size of these intervals. In statistical context, the intervals must be mutually exclusive and exhaustive. The general form of a frequency distribution is determined by the chart of normal distribution.

This statistical tool provides a visual representation for the distribution of observations within a particular analysis. The data collected in a sample is illustrated by histograms or bar graphs where the y-axis determines frequency count, and the x-axis determines the variable to be evaluated. It can also be represented in the form of a frequency table or a pie chart.

Types of Data

Now, when we focus on the types of data available for distribution, clearly there are two categories - one being grouped data and the other ungrouped data. 

Grouped data is the one where raw data is categorized into various groups represented in the form of a table. The main purpose of the table is to illustrate data points in each group. There are many ways by which a single test data set can be grouped. For example, a class of students can be grouped according to their height or the marks obtained in different subjects.

Ungrouped data, on the other hand, is raw data that has not been categorized into groups. For example, when conducting a survey where you want to analyze how many women above 50 use social media in a particular area, you first need to know the total number of women who use social media in that area.

Example: Marks of 30 students in a class in Mathematics, out of 100, are given as:

88, 90, 56, 39, 45, 60, 65, 78, 91, 99, 78, 38, 67, 85, 47, 59, 81, 33, 12, 55, 96, 23, 74, 86, 88, 69, 77, 89, 95, 91

The above data is ungrouped in nature.

On grouping, the above data can be represented as-

Calculating the Mean Deviation for Ungrouped Statistics

The simple distinction between grouped statistics and ungrouped information is that inside the case of the latter, the records are unorganized and are in random shape. This type of record is likewise referred to as raw information, whereas within the case of grouped records, it's miles organized within the shape of businesses or which has been categorized in terms of the frequency distribution. These corporations are referred to as magnificence periods.

Mean deviation determines the dispersion of all the data items in the series comparative to the measure of central tendency. This measure of central tendency is commonly median or mean. A mean deviation can also be calculated about the mode.

Let us consider a set of data that consist of observations x1, x2, x3, …………. xn

For instance, Marks of 9 college students (out of 100) are given as:

45, 60, 66, 77, 90, 37, 67, 84, 12, 

There are two steps involved in the calculation of the mean deviation of ungrouped data:

1. Calculating the Central Tendency i.e. Mean and Median

To calculate the median, the data set is to be first arranged in ascending order and the number of data present should be counted i.e. n. Depending on whether 

If n is odd, the median is \[\frac{(n+1)^{th}}{2}\]

Whereas, the Mean \[(\overline{x})\] is \[\frac{ \sum_{i=1}^{n} |x_{i}-a|}{n}\] 

2. Calculating Mean Deviation

On calculating the mean or median, the value of the central tendency, relative to which the mean deviation is calculated, is denoted by ‘a’. The absolute deviation of each data value from the measure of central tendency is being calculated first:

|x1−a|, |x2−a|, |x3−a|………..|xn−a|

Once we have all the absolute deviations, we then find out the mean of these absolute deviations for that particular set of data

Mean Absolute Deviation = \[ \frac{ \sum_{i=1}^{n} |x_{i}-a|}{n}\] 

If the central tendency measure taken into consideration is mean i.e. x, the following equation can also be adapted:

Mean Absolute Deviation = \[  \frac{\sum_{i=1}^{n} |x_{i} - \overline{x}|}{n}\] 

In the case of the median (M),

Mean Absolute Deviation = \[ \frac{\sum_{i=1}^{n} |x_{i} - {M}|}{n}\]

It is important to remember that the difference between the lowest and highest values in a set of observations, i.e. the range, is the most important factor for measuring the dispersion.

Calculating the Mean Deviation for Ungrouped data step wise

To calculate the Mean Deviation for ungrouped data, the following steps are accompanied:

let the set of data encompass  observations x1, x2, x3, …………. xn.

Step i) The degree of crucial tendency approximately which Mean Deviation is to be determined is calculated. allow this degree to be a.

Step ii) Calculate the Mean Deviation of every observation from the degree of central tendency calculated in step (i) i.e.,

x1, x2, x3, …………. xn 

Step iii) compare the mean of all the Mean Deviations. This gives the implied Mean Deviation (M.A.D) about ‘a‘ for ungrouped facts i.e.,

M. A. D (a) = \[ \frac{ \sum_{i=1}^{n} |x_{i}-a|}{n}\]

In case the measure of significant tendency means the above equation can be rewritten as:

M. A. D (x) = \[ \frac{ \sum_{i=1}^{n} |x_{i} - \overline{x}|}{n}\]

Similarly around the median,

M. A. D(a) =  \[ \frac{\sum_{i=1}^{n} |x_{i} - {M}|}{n}\]

Where as M = Median 

Did You Know?

The four steps to calculating the Mean Absolute Deviation or MAD are:

1. Find the average or mean

2. Find the value of the difference between the mean and each data point

3. For each difference, take the absolute value 

4. Find the average or the mean of the differences found