Question

What are the minimum, first quartile, median, mean, third quartile and maximum of the following data set: $10,4,9,13,5,11,5,3,7,5?$

Answer 1

Hint: We know that the minimum of a data set is the lowest number in the data set. We also know that the median is the middle number when the data contains an odd number of objects and the average of the two middle numbers when the data contains an even number of objects. Mean is the average of the objects in the data set. First quartile is the median of the lower half of the data and the third quartile is the median of the upper half of the data. And we know that the maximum of a data set is the highest number of the data.

Complete step-by-step answer:
Let us write the given data in the ascending order of numbers.
We will get $3,4,5,5,5,7,9,10,11,13.$
We know that the minimum of the above numbers is the lowest number among them.
As we can see, $3$ is the lowest number and therefore the minimum of the data set.
We know that the median of the above numbers can be found by taking average of the two middle numbers since there are $10$ numbers in the data.
So, the median is $\dfrac{5+7}{2}=\dfrac{12}{2}=6.$
We know that the first quartile is the median of the lower half of the data which lies at $25%$ of the data.
Let us write the first half of the data, we will get $3,4,5,5,5.$
Since there are $5$ numbers, we can say that the median is the middle number $5.$
We know that the mean of the data is the average of the observations and it can be found by dividing the sum of the observations by the number of observations.
So, we will get the mean as $\dfrac{3+4+5+5+5+7+9+10+11+13}{10}=\dfrac{72}{10}=7.2.$
Now, we will find the third quartile which is defined as the median of the upper half of the data which lies at $75%$ of the data.
Let us write the upper half of the data, we will get $7,9,10,11,13.$
Since there are $5$ numbers, we can say that the median is the middle number $10.$
We know that the maximum of the data set is the highest number in the data.
So, we will get the maximum of the data as $13.$
Hence the minimum is $3,$ the first quartile is $5,$ the median is $6,$ the mean is $7.2,$ the third quartile is $10$ and the maximum is $13.$

Note: We should make sure that we have written the data in the ascending order before we proceed for finding the median, first quartile and the third quartile. Because, in each of the cases, the order matters. We denote the first quartile with ${{Q}_{1}}$ and the third quartile with ${{Q}_{3}}.$