Mean of Data
Mean is the average of all the present entities in a dataset. The mean is calculated by adding all the terms of the dataset and dividing it by the number of terms. The formula for mean is given below:
Mean = \(\frac{Sum~of~the~given~data}{Total~number~of~given~data}\)
What is the ‘mean’ of a dataset?
When we have different sets of data together, then the average of these data items is known as the ‘mean’. The mean is an important value because it gives the value of center data in the given dataset.
The mean M is represented like \(\overline{M}\).
Example 1: Now, we have a table that will show the score of four terms and the mean will be calculated for these four terms:
Solution: Sum of given data = 6 + 7 + 8 + 9 = 30
Total number of terms = 4
\(\overline{M}=\frac{30}{4}=7.5\)
Example 2: In a class of 30 students, the class interval is the names of the students arranged in an ascending order and the frequency is the number of students present at that interval and we have to find the mean of the given table:
Solution: Sum of given data = 25 + 150 + 250 = 425
Sum of the frequency = 5 + 10 + 10 = 25
\(Mean=\frac{\text{Sum of the given data}}{\text{Total frequency}}\)
\(Mean=\frac{425}{25}\) = 17
The Mean Absolute Deviation or MAD is the average distance between the observation and its mean. Mean Deviation and Average Absolute Deviation is the other name for Mean Absolute Deviation.
How to calculate the mean absolute deviation?
The steps to find Mean Absolute Deviation are as follows:
Step 1: Find the average of the dataset by adding up the value and dividing it by total numbers of data.
Step 2: Now, take the value from the dataset and subtract it from the mean and leave the negative sign if you get one in the result. This is known as Absolute Deviation.
Step 3: Now, sum up the values in the above step and divide it by the sample size.
The formula for Mean Absolute Deviation is given below:
\(\text{Mean Absolute Deviation} = \frac{\sum_{}^{}\left| X-\mu \right|}{N}\)
Here, we have X as the value inside a data set, is the ‘mean’ of the given term, \(\left| X-\mu \right|\) is the absolute deviation of the data point and N is the sample size of the data set.
Example 3: The table shows the maximum speeds of eight roller coasters at an amusement park. Find the mean absolute deviation of the set of data.
Solution:
Step 1: Calculate the mean of the given data.
\(\frac{\left( 58 + 88 + 40 + 60 + 72 + 66 + 80 + 48 \right)}{8}\) = 64
The mean of the given data is 64 mph.
Step 2: Calculate the absolute value of the differences between each value in the data item and the mean.
Step 3: Calculate the mean of the absolute values of the differences between each value in the data item and the mean.
\(\frac{\left( 24 + 16 + 6 + 4 + 2 + 8 + 16 + 24 \right)}{8}\) = 12.5
The mean absolute deviation of the data provided is 12.5.
Example 4: Over a six-month period, the double bar graph depicts the monthly snowfall amounts for two cities. Take a look at the average monthly snowfalls.
Solution:
City A mean =\(\frac{\left( 3.5 + 2.2 + 1.9 + 2.1 + 2.5 + 3.4 \right)}{6}\)
= \(\frac{15.6}{6}\) = 2.6
City B mean = \(\frac{\left( 1.7 + 1.6 + 2.2 + 2.1 + 2.7 + 1.7 \right)}{6}\)
= \(\frac{12}{6}\) = 2
Because 2.6 is higher than 2, City A received more snowfall on average.
Example 5: The table displays the number of runs a pitcher has allowed in his last ten starts. Calculate the data’s mean, median, and mean absolute deviation.
Order the Run allowed: 0, 0, 0, 2, 4, 4, 5, 6, 6, 8
Mean = \(\frac{35}{10}\) = 3.5
Median = \(\frac{(4+4)}{2}\) = 4
Mean Absolute Deviation
\(\text{Mean Absolute Deviation} = \frac{\sum_{}^{}\left| X-\mu \right|}{N}\)
Mean Absolute Deviation = \(\frac{24}{10}\) = 2.4
The ratio of the sum of all observations to the total number of observations in a data set is known as the mean in statistics.
Mean = (2 + 6 + 4 + 5 + 8) / 5
= 25 / 5 = 5
When a data set is arranged in order, the When a data set is arranged in order, the median is the most central value.
Arrange the data in ascending order as follows:
1,2,3,4,7,8,10
4 is the median (middle) value.