Question

Observations of some data are \(\frac{x}{5}, x, \frac{x}{3}, \frac{2 x}{3}, \frac{x}{4}, \frac{2 x}{5}\) and \(\frac{3 x}{4}\) where \(x>0\). If the median of the data is 4, then find the value of \(^{\prime} \mathrm{x}\) '. (1) 5 (2) 7 (3) 8 (4) 10

Short answer

**Answer:** (4) 10

Step-by-step solution

Arrange the observations in ascending order

First, we rewrite the observations with their respective coefficient: \(\frac{1}{5}x, x, \frac{1}{3}x, \frac{2}{3}x, \frac{1}{4}x, \frac{2}{5}x, \frac{3}{4}x\). Then, we arrange the observations in ascending order: \(\frac{1}{5}x, \frac{1}{4}x, \frac{1}{3}x, \frac{2}{5}x, \frac{2}{3}x, \frac{3}{4}x, x\).

Find the median and solve for x

Since there are 7 observations, the median is the 4th value. Therefore, \(\frac{2}{5}x = 4\). To solve for x, we can multiply both sides by \(\frac{5}{2}\): $$x = 4\cdot \frac{5}{2}.$$

Calculate x

Now, let's perform the multiplication: $$x = 4\cdot \frac{5}{2} = \frac{4\cdot 5}{2} = \frac{20}{2} = 10$$ So, the value of x is 10. Checking the options, we find that the correct option is: (4) 10

Key concepts

Observations in Statistics

In statistics, observations refer to the recorded values from a dataset, which in this exercise are given as expressions involving a variable, \(x\). When working with observations, it's important to understand that each value represents a specific element in the data set that contributes to the statistical analysis.
Our data set consists of values based on \(x\), such as \(\frac{x}{5}, x, \frac{x}{3}, \text{etc.}\). These individual observations help us analyze and find the median. Observations can be numbers, or like in this case, expressions involving a variable.

• An observation could be anything that you can measure or record.
• In this scenario, observations are presented as fractions of \(x\).
• Understanding each observation helps when calculating statistical measures like the median.

Observations are the foundation of data analysis, and they help provide insights when used correctly.
When arranged properly, these observations can reveal meaningful information about the distribution of data in a set.

Order of Values

When dealing with a collection of observations, one crucial step is organizing them in ascending or descending order. This lets us analyze characteristics like the median or mode more accurately. To find the median in statistics, ordering the data is necessary.
In the given exercise, ordering the observations helps us find the median efficiently. Start with the smallest value and proceed to the largest:

• \(\frac{1}{5}x\)
• \(\frac{1}{4}x\)
• \(\frac{1}{3}x\)
• \(\frac{2}{5}x\)
• \(\frac{2}{3}x\)
• \(\frac{3}{4}x\)
• \(x\)

Sorting allows us to determine that the 4th value, \(\frac{2}{5}x\), is the median if there are seven observations, providing a clear path to solve for \(x\).
This methodical ordering helps ensure accuracy when computing statistical measures and offers better insight into the dataset's distribution.

Arithmetic Operations

Performing arithmetic operations is essential when handling statistical problems or solving mathematical equations that arise from them.
In our exercise, we identify the median and set this observation equal to 4, enabling us to solve for \(x\). We do this by performing multiplication, a basic arithmetic operation. Let's break down the steps:

• We have \(\frac{2}{5}x = 4\).
• To isolate \(x\), we multiply both sides of the equation by \(\frac{5}{2}\).
• This is because multiplying by \(\frac{5}{2}\) is the inverse operation of \(\frac{2}{5}\), which effectively cancels it out.

This gives us \(x = 4 \cdot \frac{5}{2}\), simplifying to \(x = 10\).
Arithmetic operations, like multiplication and division, allow us to manipulate and solve equations derived from statistical observations. They form the core of unlocking the meanings hidden within the data.