Question

What is the mean, median, mode and range of 2, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9?

Answer 1

Hint: We know that the mean is the ratio of total sum and total number of values. Also, the median is the middlemost value in an arranged data set. We know that mode is that value which has the highest frequency, and the range is the difference between the largest and smallest value.

Complete step-by-step answer:
We are given with the following dataset: 2, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9.
Let us first count the number of values in the given dataset. We can clearly see that we have 19 values.
Thus, N = 19.
We know that the mean or average is the ratio of sum of all the values and the total number of values. So, let us add all the values and calculate the sum.
Sum = 2 + 3 + 3 + 3 + 3 + 4 + 4 + 5 + 6 + 6 + 6 + 6 + 7 + 7 + 8 + 8 + 8 + 8 + 9
Thus, Sum = 106.
We know that the total number of values, N = 19.
Thus, we have
Mean = $\dfrac{\text{Sum of all values}}{\text{No}\text{. of values}}$
And so, Mean = $\dfrac{106}{19}\approx 5.58$.
We know that in an arranged dataset, median is the centremost value. Here, we already have an arranged data set. Also, N is odd.
So, the Median term = $\left( \dfrac{N+1}{2} \right)$th term.
Thus, the Median term = $\left( \dfrac{19+1}{2} \right)$th term = $10$th term.
We can see that the 10th term is 6. Thus, median = 6.
We know that the value with the highest frequency is called the mode.
Here, we can see that the values 3, 6 and 8 have the highest frequency, and all of these are repeated three times.
Thus, the modes are 3, 6 and 8.
We also know that the difference between the highest and lowest value is called the range.
Here, we can see that the highest value is 9, and the lowest value is 2.
Thus, range = 9 – 2 = 7.
Hence, mean is 5.58, median is 6, modes are 3, 6 and 8, and the range is 7.

Note: We must take care that median is always calculated for a data set arranged in either increasing or decreasing order. We must also note that this is a tri-modal distribution. We must also take care while counting the number of values.