# **Absolute Mean Deviation of Ungrouped Data**

There are various techniques to measure how closely your data values are grouped while working with data. The Mean deviation is the most common. Most individuals learn how to calculate using the Mean deviation calculator in school by adding all the data values together and dividing by the number of items in the set. However, the mean deviation around the mean is a more complex calculation. This computation determines how near your numbers are to the mean. Finding this consists of finding the mean for a data set, finding the difference of each data point from that mean, and then taking the mean of those differences.

The absolute mean deviation of data points from a centre point is absolute deviation. The mean, median, mode, or random point can be used as the centre point. The mean is frequently used as the axis. The absolute mean deviation formula is used to calculate the mean absolute deviation (MAD), the average of the data points’ absolute deviations (distances) from the data set’s mean. Compared to the central tendency metric, mean deviation determines the dispersion of all data items in the series. The median or mean is a popular metric of central tendency. You can calculate the mean deviation for both grouped and ungrouped data.

In this article, we will be exploring the Absolute Mean Deviation of Ungrouped Data.

**What is the mean deviation?**

A measure of central tendency is the mean deviation. We can figure it out using the Arithmetic Mean, Median, or Mode. It demonstrates how distant all of the observations are, on average, from the middle. Because it is an absolute value, each deviation is an absolute deviation, ignoring the negative signs. In addition, the deviations on both sides of the mean must be the same.

## What is ungrouped data?

The primary distinction between grouped and ungrouped data is that ungrouped data is unstructured and in a random format. This sort of data is also known as raw data, whereas grouped data is data that has been arranged into groups or classed according to the frequency distribution. Class intervals are the names given to these groups.

Ungrouped data is a sort of distribution in which each piece of information is shown in its raw form. For instance, a student’s last five test results are 65, 94, 85, 77, and 80. We can conclude her performance by deducing the range and mean deviation from this data.

### What are the range and its deviation?

The range is the difference between the distribution’s highest and minimum values. The sum of absolute dispersion values to the number of observations is technically defined as mean deviation.

How to calculate the Absolute Mean Deviation of Ungrouped Data with a formula?

The sample mean is used to calculate the mean deviation for ungrouped data. First, the difference between each item in the distribution (data set) and the mean is determined by absolute value. Then, subtract the mean from each number in the data collection, ignoring the positive and negative signs (consider everything to be positive).

Finally, the sum of all the differences is divided by the number of items in the sample.

The mean deviation is the average of the final departures from a reasonable average of the observations or values. The appropriate average could be the mean, median, or mode. The mean absolute deviation is another name for it. We’ll learn more about specific vital formulas today in this paragraph, such as the mean deviation formula for a discrete or continuous series, and so on.

** This is the formula that must be used:**

Where X is the value of the observation and y is the population means.

N is the total number of observations in the sample. X is the sample mean.

The letter n denotes the number of observations in the sample.

Steps that you should follow to derive the absolute mean deviation of ungrouped data

The average distance between each data value and the mean is the mean absolute deviation (MAD) of a data set. A measure of variance in a data set is the mean absolute deviation. The mean absolute deviation tells us how “scattered” the values in a data set are.

The following steps are used to compute the mean deviation for ungrouped data:

Allow observations to make up the data set: Let the set of data consist of observations x 1, x2, x 3 … … … . . x n.

Step i) Determine the measure of central tendency used to calculate the mean deviation. Assume that this is ‘a’.

Step ii) From the measure of central tendency computed in step, I calculate the absolute deviation of each observation, i.e., |x1−a|,|x2−a|,|x3−a|………..|xn−a|

Step iii) Compute the average of all absolute variances. This yields the mean absolute deviation (M.A.D.) around ‘a’ for ungrouped data. Where, M.A.D (a) = ∑ n i = | x i – a |/ n

** If mean is used as a measure:**

If mean is used as the measure of central tendency, the equation above may be rewritten as:

M.A.D(¯x)=∑ni=|xi–¯x|/n

Where¯x=Mean

If Median is used as a measure:

Around the median, it’s similar:

M.A.D(a) = ∑ n i = | x i – M |/ n

M is the median value.

**Let us look at this example to understand this better:**

Arjun enjoys posting videos of his guitar online. Here’s how many “likes” the past five videos received: 10, 15, 15, 17, 18, 21. To find the Absolute Mean Deviation, we need to calculate the mean.

Hence, the mean will be: The sum of the data is 96 total “likes,” and there are six pictures.

Which is 96/6 = 16.

The distance between each data point and the mean should now be calculated.

| 10 – 16 | = 6

| 15 – 16 | = 1

| 15 – 16 |= 1

|17 – 16 |= 1

| 18 – 16 | = 2

| 21 – 16 | = 5

Now we add these numbers together and divide it by number of data points: 6 + 1 + 1 + 1 + 2 + 5 = 16 / 6 = 2.67

**Conclusion:**

The absolute mean deviation is one of the best ways to understand and analyze a set of data because it is simple to calculate, it is based on all of a series’ observations, it shows the dispersion, or scatter, of the various items of a series from its central value, and it is unaffected by the values of the series’ extreme items. We hope this article has given you a better understanding of mean deviation.