Question

The mean of 100 items is 48 and their standard deviation is 10. Find the sum of all the items.

Answer 1

Hint: Use the mean formula which is the average of the sum of the terms and is calculated by $\bar x = \dfrac{{\sum {{x_i}} }}{n}$. Then substitute the mean of the items and the total number of items. Then, do the simplification to get the sum of all items.

Complete step-by-step solution:
The mean of 100 items is 48 and their standard deviation is 10.
We have to find the sum of all items.
Mean is the measure of the average of a set of values which can be calculated by dividing the sum of all the observations by the number of observations. There are four different ways to measure the mean of a data set. They are arithmetic mean, geometric mean, harmonic mean, and weighted arithmetic mean. Usually, the arithmetic mean is calculated because it is easy to calculate.
The mean is calculated by dividing the sum of observations by the number of observations. The formula is given by,
$\bar x = \dfrac{{\sum {{x_i}} }}{n}$
Now, substitute the value of mean and number of items in the mean formula,
$ \Rightarrow 48 = \dfrac{{\sum {{x_i}} }}{{100}}$
Cross-multiply the terms,
$\therefore \sum {{x_i}} = 4800$

Hence, the sum of all the items is 4800.

Note: Whenever we face such types of problems the key point that we need to recall is that Mean or Arithmetic mean is the average of given numbers. It is found by calculating the sum of the given numbers and dividing it by how many numbers there are. These types of questions are based only on this definition and can be easily solved using this.