4.4.3 Averages from Grouped Data
Grouped data:
Grouped data is a type of data that has been organized into groups or classes.
Each group or class represents a range of values that fall within that group.
Grouped data is commonly used in statistical analysis when the data is too large or too varied to be presented in individual values.
Grouped data is usually presented in a frequency distribution table, which shows the number of observations that fall within each group or class.
For example, a frequency distribution table might show the number of students in a school who scored within certain ranges on a test, such as \( 0-10 \), \( 11-20 \), \( 21-30 \), and so on.
Mean from grouped data:
To find the average (or mean) from grouped data follow these steps
Write down the frequency distribution table that contains the class intervals and their corresponding frequencies.
Find the midpoint of each class interval.
To do this, add the lower and upper limits of each class interval and divide the sum by \( 2 \).
The formula is:
\( midpoint = \frac{lower \ limit+upper \ limit}{2} \)
Multiply each midpoint by its corresponding frequency.
Add up the products obtained in step (iii).
Add up the frequencies from the frequency distribution table.
Divide the sum obtained in step (iv) by the sum obtained in step (v).
The result is the average (or mean) from grouped data.
Worked example:
Suppose you have the following frequency distribution table:
To find the average from this data
Write down the frequency distribution table.
Find the midpoint of each class interval:
Multiply each midpoint by its corresponding frequency:
Add up the products obtained in \( 4^{th} \) column
\( 75+300+700+360+275=1710 \)
Add up the frequencies
\( 5+12+20+8+ 5=50 \)
Divide the sum from \( 4^{th} \) column by the sum of frequencies
\( \frac{1710}{50}=34.2 \)
Therefore, the average (or mean) from the grouped data is \( 34.2 \).
Median from grouped data:
To find the class interval that the median lies in from grouped data
Find the cumulative frequency of the data.
This can be done by adding up the frequencies of each class interval, starting from the first-class interval.
Determine the median value of the data.
If the total number of observations (i.e., the sum of the frequencies) is odd, the median is the value that is exactly in the middle of the dataset.
If the total number of observations is even, the median is the average of the two middle values.
Identify the class interval that contains the median value.
To do this, compare the cumulative frequency of each class interval with the median value.
The class interval that contains the median value is the one for which the cumulative frequency is just greater than or equal to the median value.
Worked example
Suppose you have the following frequency distribution table
Find the cumulative frequency by adding up the frequencies of each class interval,
starting from the first-class interval:
Determine the median value of the data.
In this case, the total number of observations is \( 60 \), which is even.
So, the median is the average of the two middle values, which are the \( 30^{th} \) and \( 31^{st} \) values in the dataset.
To find these values, use the cumulative frequency.
The \( 30^{th} \) value falls within the class interval \( 10 \) -\( 15 \).
The lower limit of this interval is \( 10 \), and the frequency is \( 20 \).
So, the 30th value is greater than or equal to \( 10 \), and less than \( 15 \).
Modal class interval:
To find the modal class interval from grouped data
Determine the frequency of each class interval in the grouped data.
Identify the class interval with the highest frequency.
This is your modal class interval.
If there are two or more class intervals with the same highest frequency, then you can consider them all as modal class intervals.
Worked example:
Determine the frequency of each class interval in the grouped data.
The frequency of each class interval is given in the second column of the table.
Identify the class interval with the highest frequency. This is your modal class interval.
The class interval with the highest frequency is \( 30 \) – \( 40 \), with a frequency of \( 18 \).
Therefore, the modal class interval is \( 30 \) – \( 40 \).
Range
Range and interquartile range are both measures of spread or variability in a dataset. However, they have different methods of calculation and provide different information about the distribution of the data.
Range:
The range is a measure of the spread of a dataset, calculated as the difference between the highest and lowest values in the dataset.
For example, if you have a dataset of exam scores ranging from \( 60 \) to \( 95 \), the range would be \( 35 \) \( (95 \) – \( 60) \).
The range is a simple measure of spread that provides an idea of how much the data varies from the minimum to the maximum value.
Worked example:
Suppose we have the following dataset:
\( {1, 5, 10, 15, 20} \)
To calculate the range, we subtract the minimum value \( (1) \) from the maximum value \( (20) \), which gives us a range
\( Range=20-1=19 \)
So the range of this dataset is \( 19 \).
Interquartile Range (IQR):
The interquartile range is a measure of spread that is less sensitive to outliers than the range.
It is calculated as the difference between the upper and lower quartiles of the dataset.
The lower quartile is the value that cuts off the lowest \( 25 \)% of the data, and the upper quartile is the value that cuts off the highest \( 25 \)% of the data.
The interquartile range contains the middle \( 50 \)% of the data.
For example, if the lower quartile is \( 25 \) and the upper quartile is \( 75 \), the \( IQR \) would be \( 50 (75 \) – \( 25) \).
The \( IQR \) provides information about the spread of the data in the middle half of the distribution, which is useful for identifying any potential outliers or skewness in the data.
Worked example:
Suppose we have the following dataset
\( {7, 12, 18, 23, 29, 31, 36, 42, 46, 51} \)
To calculate the quartiles, we first need to order the data from smallest to largest
\( {7, 12, 18, 23, 29, 31, 36, 42, 46, 51} \)
The median \( (Q_2) \) is the middle value,
\( Q_2 =29 \)
To find \( (Q_1) \), we need to take the median of the lower half of the dataset (excluding \( 29 \))
So the lower half is \( 7, 12, 18, 23 \) and the median of that is \( 15.5 \) \( Q_1=15.5 \)
To find \( (Q_1) \), we need to take the median of the upper half of the dataset (excluding \( 29 \))
So the upper half is \( {31, 36, 42, 46, 51} \), and the median of that is \( 42 \)
\( Q_3=42 \)
So the quartiles for this dataset are \( Q_1= 15.5 \), \( Q_2= 29 \), \( Q_3= 42 \)
The interquartile range \( IQR \)is the difference between \( (Q_3) \) and \( (Q_1) \), which is \( 26.5 \).