The aim of this question is to understand the key concepts of **mean** and **median** that form the basis of statistical calculations.

The **mean** of a given sample of data is defined as the **average numerical value** (or **arithmetic mean**) of all the values. Mathematically:

\[ Mean \ = \ \dfrac{ \text{ sum of all values of the sample data } }{ \text{ total no. of samples } } \]

\[ \Rightarrow Mean \ = \ \dfrac{ x_1 \ + \ x_2 \ + \ x_3 \ + \ … \ + \ x_n }{ n } \]

Where $ x_1, \ x_2, \ x_3, \ … \ , \ x_5 $ are the **values of sample data** and $ n $ is the **total no. of samples** or sample size.

Mean can be **used to calculate** important statistical characteristics of data such as **variance**, **standard deviation**, and other **moments** / **central moments**.

The **median** of a given sample of data is an **order property**. It is defined as the **middle value** of all the values given in the sample after **sorting all values in ascending order**. Mathematically:

\[ Median \ = \ \left \{ \begin{array}{ll} X[ \frac{ n }{ 2 } ] & \text{ if n is odd } \\ \dfrac{ X[ \frac{ n \ – \ 1 }{ 2 } ] \ + \ X[ \frac{ n \ + \ 1 }{ 2 } ] }{ 2 } & \text{ if n is even } \end{array} \right. \]

Where $ X $ is the ordered list of **sample values** and $ n $ is the **total no. of samples** or sample size.

## Expert Answer

In the given question, the **company’s stance** is that the **average value of absences per employee is 7 days**. They are actually talking about the **sample mean** here. They have summed the **total no. of leaves of all employees** and divided it by the **total no. of employees**.

The **union negotiator’s stance** is that the **average employee** takes a maximum leave of 3 days. They are actually talking about the **median of the same data**.

**Both** the company and union have the **correct figures** but their point of view is different. **Statistically**, the company is talking about **the mean** whereas the union negotiators are considering **the median**.

## Numerical Result

Both are correct.

\[ Mean \ = \ 7 \ days \]

\[ Median \ = \ 3 \ days \]

## Example

Let’s say that for a given company, there are **9 employees**. Here are the **leaves taken in the past year**:

\[ \{ \ 1, \ 2, \ 4, \ 6, \ 0, \ 2, \ 9, \ 1, \ 20 \ \} \]

Calculate the **mean and median** of the sample data.

\[ \Rightarrow Mean \ = \ \dfrac{ 1 + 2 + 4 + 6 + 0 + 2 + 9 + 1 + 20 }{ 10 } \ = \ \dfrac{ 45 }{ 9 } \ = \ 5 \ days\]

**Sorting the given data in ascending order:**

\[ \{ \ 0, \ 1, \ 1, \ 2, \ \boldsymbol{ 2 }, \ 4, \ 6, \ 9, \ 20 \ \} \]

\[ Median \ = \ \text{ Middle Value } \ = \ \text{ 5th value } \ = \ 2 \ days \]