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A median is a 'half-way' point in our data. We can use quartiles and percentiles to find any fraction (or percentage) of the way into our data.

Quartiles divide a data set into four equal parts. The three quartiles are:

**Lower Quartile (Q**: The value below which 25% of the data falls. This is also known as the 25th percentile._{1})**Median (Q**: The middle value of the data set, also known as the 50th percentile._{2})**Upper Quartile (Q**: The value below which 75% of the data falls, also known as the 75th percentile._{3})

To find the quartiles for a small discrete data set, use these formulas for the positions of the quartiles:

\[Q_1 = \frac{n}{4}\]

\[Q_2 = \frac{n+1}{2}\]

\[Q_3 = \frac{3n}{4}\]

Where \(n\) is the total number of data points. With \(Q_1\) and \(Q_3\), we always round this number up, unless it is a integer where we add 0.5.

To find the quartiles for a large grouped data set, use these formulas for the positions of the quartiles:

\[Q_1 = \frac{n}{4}\]

\[Q_2 = \frac{n}{2}\]

\[Q_3 = \frac{3n}{4}\]

We use the exact values in our further calculations.

Percentiles divide a data set into 100 equal parts. The **k-th percentile** is the value below which \(k\%\) of the data falls. The formula for the position of the k-th percentile is:

To find a percentile for a large grouped data set, use these formulas for the position of the k-th percentile:

\[P_k = \frac{kn}{100}\]

Where:

- \(P_k\) is the k-th percentile.
- \(k\) is the desired percentile (e.g., 20 for the 20th percentile).
- \(n\) is the number of data points.

When the data is grouped and we have the position of the quartile or percentile, we use **linear interpolation** to estimate the exact value. This method assumes the data is evenly distributed within the class interval.

For interpolation, the formula is:

\[\text{Estimated Value} = L + \left( \frac{P - F}{f} \right) \times h\]

Where:

- \(L\) is the lower boundary of the class interval containing the quartile or percentile.
- \(P\) is the position of the quartile or percentile you are calculating.
- \(F\) is the cumulative frequency before the class.
- \(f\) is the frequency of the class interval containing the quartile or percentile.
- \(h\) is the width of the class interval.

Suppose you want to calculate the 30th percentile, and the 30th percentile falls within a class interval with:

- Lower boundary \(L = 10\)
- Class width \(h = 5\)
- Cumulative frequency before the class \(F = 20\)
- Frequency of the class \(f = 8\)
- The position \(P = 30\)

The estimated value would be:

\[

\text{Estimated Value} = 10 + \left( \frac{30 - 20}{8} \right) \times 5 = 10 + \left( \frac{10}{8} \right) \times 5 = 10 + 6.25 = 16.25

\]

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