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The average deviation formula is used to characterize the dispersion among the given set of data. The average deviation is the average deviation of each observation from the mean value of the observation. The mean of the observations is calculated before the average deviation is calculated. The average deviation formula becomes a handy tool to quickly calculate the average deviation and reach the result very quickly. Let us study the average deviation formula using solved examples.
What Is Average Deviation Formula?
The average deviation formula is used to calculate the average deviation of the observation from the mean value of the observation. The average deviation formula for n number of observations is given below:
- x̅ is the mean.
- n is the total number of data values
Let’s take a quick look at a couple of examples to understand the average deviation formula, better.
How To Calculate Average Deviation
Calculating the average deviation can be an effective way to analyze the variability within a data set. Regardless of the exact nature of the compiled data, knowing its average deviation can help you interpret it. Knowing how to calculate the average deviation is a valuable skill, but it takes study and practice. In this article, we discuss what average deviation is, how to calculate it, as well as the differences between absolute and average deviation, mean average and average deviation from the mean and standard deviation versus average deviation.
Consider these steps when calculating the average deviation of a data set:
1. Calculate the mean/median
The first step is calculating the mean. You can do that by adding all the values in the data set and dividing the resulted sum by the total number of values.
Alternatively, you can calculate the median if you wish to use it instead of the mean. Arrange all numbers in numerical order and count how many there are in total. Then, if the total number is an odd one, divide it by two and round up to find the position of the median. If the total number is even, divide it by two and make an average between the number in that position and the one in the next higher position.
2. Calculate the deviation from the mean
After calculating the mean, you can calculate the deviation from the mean for each value in the data set. Calculate the difference between the previously calculated mean and each value in the data set and write down the absolute value of the resulting numbers. The absolute value of a number is its modulus or non-negative value. Since the direction of each variation is irrelevant when calculating the average deviation, all resulting numbers are positive.
3. Calculate the sum of all deviations
After calculating the deviation from the mean for each value in the data set, you need to add them together. Since this is an absolute value operation, each value should be a positive number.
4. Calculate average deviation
Finally, calculate the average deviation of your data set by dividing the previously calculated sum of all deviations by the total number of deviations that you added together. The resulting number is the average deviation from the mean.
Example
Consider this example when calculating the average deviation from the mean.
A basketball player played 5 games so far this season. The scoring numbers from each game are 23, 30, 31, 15 and 46.
The first step is calculating the mean. You do this by adding the points and dividing the result by the five games.
23+30+31+15+46=145
145/5=29
Now that you determined that the player has scored an average of 29 points per game, you need to calculate the deviation from the mean for each game.
23-29=6
30-29=1
31-29=2
15-29=14
46-29=17
Next, you need to calculate the sum of all variations.
6+1+2+14+17=40
The average deviation is the sum of all deviations divided by the total number of entries.
Average deviation=40/5=8
The average deviation from the mean regarding the points scored in the first five games of the season is 8.
Examples on Average Deviation Formula
Example 1: Find the average deviation of the given data using the average deviation formula: 12, 14, 16, 18, 20, 22.
Solution:
To find: average deviation
Given: n = 6
Finding mean of the given data:
x̅ = (12 +14 + 16 + 18 + 20 + 22)/6 = 17
Using the average deviation formula
Average deviation = (|12 – 17| + |14 – 17| + |16 – 17| + |18 – 17| + |20 – 17| + |22 – 17|)/6
= (5 + 3 + 1 + 1 + 3 + 5)/6 = 18/6 = 3
Average deviation = 3
Answer: Hence the average deviation of the given data is 3.
Example 2: Find the average deviation of the given data using the average deviation formula: 33, 44, 55, 66, 77, 88, 99.
Solution:
To find: average deviation
Given: n = 7
Goinding mean of the given data:
x̅ = (33 + 44 + 55 + 66 + 77 + 88 + 99)/7 = 66
Using the average deviation formula,
Average deviation = (|33 – 66| + |44 – 66| + |55 – 66| + |66 – 66| + |77 – 66| + |88 – 66| + |99 – 66|)/7
Average deviation = 18.857
Answer: Hence the average deviation of the given data is 18.857.
Example 3: A football player played 5 games so far this season. The scoring numbers from each game are 14, 10, 9, 7, and 5. Determine the mean and calculate the average deviation.
Solution:The average of the score is = (14 + 10 + 9 + 7 + 5)/5 = 45/5 = 9
the average score = 9
To calculate the average deviation we need to calculate the deviation from the average for each game.
|14 – 9| = 5
|10 – 9| = 1
|9 – 9| = 0
|7 – 9| = 2
|5 – 9| = 4
Sum of variations = 5 + 1 + 0 + 2 + 4 = 12
Average deviation = 12/5 = 2.4
Answer: Hence the average deviation of the given data is 2.4
Question 1: Calculate the average deviation for the given data: 4, 6, 8, 10, 12, 14.
Solution:
Given:
n = 6
First lets find the mean by using the formula,
Question 2: Find the average deviation for the following set of observations.
11, 6, 6, 12, 12, 7, 7, 9
Solution:
Given set of observations:
11, 6, 6, 12, 12, 7, 7, 9
Sum of observations = 11 + 6 + 6 + 12 + 12 + 7 + 7 + 9 = 70
Number of observations = n = 8
Mean = Sum of observations/Total number of observations
Tips for calculating average deviation
Here are some tips to help you calculate average deviation effectively:
Write the equation down
When calculating average deviation, there are steps that involve calculating a data set’s set central point, which is the mean or median, then calculating the deviation from it. You might write the equation down and complete the proper steps in a written format, rather than using a calculator. This may help you remember each step of the equation, and you can refer back to your previous calculations to assist you with future calculations. If you encounter obstacles when finding the average deviation, then you may review your written steps to identify an error or miscalculation.
Use calculation software
You might use calculation software to work with large data sets, since it may be challenging to write out data sets that have a large amount of variables. Typically, you may write out data sets that have between five and 10 variables. When a data set has over 10 variables, you might use a calculator or input the information into calculation software. Doing so may also help you verify that you have the correct average deviation.
If using a calculator, you may have to record the answers on a sheet of paper to complete your calculation, since some calculators perform basic math operations. You may find a calculator that has software that allows you to input the variables from a data set into the device so that it can perform the calculations.
FAQs on Average Deviation Formula
How To Calculate the Mean of Data in Average Deviation Formula?
In the average deviation formula, the very first thing we calculate is the average of the given data set by dividing the sum of observations by a total number of observations.
How To Calculate the xi – x̅ in the Average Deviation Formula?
In the average deviation formula, the term (xi – x̅) is necessary to calculate as it gives the final value that is the average deviation of the given data. to calculate (xi – x̅) we first calculate the average of the given data then individually by subtracting the mean value from each observation the value of (xi – x̅) can be evaluated.
What Are the Steps To Calculate the Average Deviation Using Average Deviation Formula?
While calculating the average deviation using the average deviation formula we should follow the steps given below:
- Calculate the average of the data observations given.
- Calculate the value of (xi – x̅) deviation from the average.
- Calculate the sum of all deviations (variations).
- Calculate average deviation that is the ratio of the sum of variations to the total number of observations.
What is the average deviation?
Average deviation is a statistical tool that provides the average of different variations from a data set. The purpose of average deviation is to measure the distance of a deviation from the data set’s mean or median. The mean value is the average of all numbers within a data set, while the median is the value of the number found in the middle of the data set after arranging each data set value in order from lowest to highest. Math professionals may refer to the average deviation as the mean absolute deviation (MAD) or the average absolute deviation.
What does average deviation tell you?
The average deviation shows you the distance that one specific variable is from the mean of a data set. This can help you analyze information depending on your needs. Many data scientists, market researchers and data analysts use average deviation to determine how far a variable deviates from a large data set’s mean.
This may help them analyze data more quickly and easily. For example, if a data scientist is measuring a data set that indicates the different types of weather in the summer, they may use average deviation to determine how much each day’s forecast deviates from the average temperature and weather conditions for the summer.
Why is average deviation important?
Here are some reasons why average deviation is important:
- Assessing variability: You may use average deviation to determine if your data set has a high degree of variability or if the data is within a specific range.
- Interpreting the average: After calculating the average deviation, you can interpret the data to make changes or set goals. For example, if a student calculates the average deviation of their test scores, they may determine that the average deviation of their first 10 test scores of the semester is 25%, and they may set a goal to maintain more consistent test scores that have a lower deviation.
- Showing trends: Average deviation may show trends in a data set, like if there is a large cluster of data that is close to the average deviation, or if data is far from the mean.
Absolute deviation vs. average deviation
Calculating the absolute deviation is a crucial step for determining what the average deviation is. The absolute deviation is the difference between a data set’s mean and each value in the respective data set. The name of absolute deviation comes from the fact that the resulting numbers are all written down as absolute numbers. The measure expresses the distance between the mean and each value, therefore it being a negative or positive number is irrelevant.
After calculating the absolute deviation for each value in the data set, you can calculate the average deviation by adding them all together and dividing them by the total number of values in the data set.
Mean average vs. average deviation from the mean
Calculating the mean average is also a crucial step in determining what the average deviation from the mean is. The mean average is simply the sum of all values included in the data set, divided by the total number of values. Calculating the mean average helps you determine the deviation from the mean by calculating the difference between the mean and each value. Next, divide the sum of all previously calculated values by the number of deviations added together and the result is the average deviation from the mean.
Standard deviation vs. average deviation
Standard deviation is also a measure of variability within a data set, as it shows the size of deviation between all values in the data set. The main difference between the two is that the resulting values from subtracting the mean from the value of each data point are only written as absolutes when calculating the average deviation. To calculate standard deviation, the resulting values are not written in absolutes, but squared. Then, you need to calculate the mean of all the squared values. The square root of that mean is the standard mean.
Standard deviation is more commonly used to measure variability, being a very popular tool to calculate the volatility of financial instruments and potential investment returns. Higher volatility typically means that there is an increased risk of an investment generating a loss, meaning that an investor that takes on the risk of high-volatility security typically expects a high return from it. The average deviation is also used as a financial tool, but typically less often than the standard deviation.
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