Question

If the mean of \(26,19,15,24\) and \(x\) is \(x\), then find the median of the data. (1) 23 (2) 22 (3) 20 (4) 21

Short answer

Answer: The median of the given data set is 21.

Step-by-step solution

Write down the formula for the mean and substitute the given values

The formula for the mean of a set of numbers is: Mean = (Sum of all numbers) / (Total number of values in the set) We are given the numbers \(26, 19, 15, 24\) and \(x\). The mean of these five numbers is equal to \(x\). We can write the equation as follows: $$x = \frac{26 + 19 + 15 + 24 + x}{5}$$

Solve for x

Now let's solve the equation for x: $$5x = 26 + 19 + 15 + 24 + x$$ $$4x = 26 + 19 + 15 + 24$$ $$4x = 84$$ $$x = 21$$

List the numbers in ascending order

Now that we have the value of x, we can list the given numbers and x in ascending order. This step is necessary to find the median: \(15, 19, 21, 24, 26\)

Find the median of the data

The median is the middle value of a set of numbers when they are listed in ascending order. In this case, we have 5 numbers, so the median is the 3rd value in the list (the middle value): \(15, 19, \bold{21}, 24, 26\) The median of the data is \(\boxed{21}\), which corresponds to option (4).

Key concepts

Mean Formula

Understanding the mean of a set of numbers is crucial when dealing with statistical data. The mean, often known as the average, is calculated by using the mean formula, which is summing up all the values in a set and then dividing that sum by the number of values.

For instance, if you're given a series of numbers such as 26, 19, 15, 24, and an unknown value represented by 'x,' the mean would be represented as:$$ \text{Mean} = \frac{26 + 19 + 15 + 24 + x}{5} $$
In this equation, you're adding up all the numbers and then dividing by 5 because there are 5 values in total, including 'x.' The mean is a measure of central tendency and gives you a sense of what's 'typical' for that dataset, balancing out extreme values that could skew the perspective.

Solving for x in Mean Equation

When you're given a mean equation with an unknown variable, solving for 'x' can be a bit like detective work! Here's the gist of it: You're told that the mean of a set of numbers is equal to one of the numbers itself. This is an intriguing setup because it gives you both the tool (mean formula) and a piece of the puzzle (the mean is one of the numbers), letting you crack the code.

Using the same numbers as before – 26, 19, 15, 24, and 'x' with the mean being 'x', setting up the equation looks like this:$$x = \frac{26 + 19 + 15 + 24 + x}{5}$$
To find 'x', multiply both sides by 5 to eliminate the fraction, then subtract 'x' from both sides to get '4x' on one side. Solve for 'x' by dividing the sum of the numbers by 4. It's like unraveling a math-laden mystery, and at the end, the value you unearth for 'x' becomes a piece in solving other related problems, like finding the median.

Ordering Data to Find Median

The median is the middle value of a data set when it's ordered from smallest to largest, or vice versa. The process is straightforward but crucial – you have to line up all your numbers in ascending order. If you have an odd number of values, like in our example of 5 numbers, the median will be the one right in the center.

Once you've ordered the data – 15, 19, 21, 24, 26 – it's simply a matter of pinpointing that central number. With an even number of values, you'd take the two middle values and calculate their mean to find the median. It’s essential to order the data correctly as an initial step, because if you mix up the order, the median you identify could lead to an inaccurate picture of your dataset.

By finding the value of 'x', you've filled in the missing piece and can now ensure the data is ready for you to locate the median, which in this case turned out to be 21. This step is a blend of organization and calculation, fundamental for correct data analysis.