PERCENTILE WORKSHEET (LESSON 5 | STATISTICS) | Study Guides ... - Free Printable

Problem Analysis:

The task involves calculating percentiles for a given dataset. The dataset provided is:

$$
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
$$

We are asked to find the following percentiles:
1. 25th Percentile
2. 50th Percentile
3. 75th Percentile

Step-by-Step Solution:

#### Step 1: Understand Percentiles
A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. For example:
- The 25th percentile (also called the first quartile, \( Q_1 \)) is the value below which 25% of the data falls.
- The 50th percentile (also called the median, \( Q_2 \)) is the value below which 50% of the data falls.
- The 75th percentile (also called the third quartile, \( Q_3 \)) is the value below which 75% of the data falls.

#### Step 2: Formula for Percentile Calculation
The formula to find the position of the \( P \)-th percentile in an ordered dataset of size \( n \) is:

$$
L = \frac{P}{100} \times (n + 1)
$$

Where:
- \( L \) is the location of the percentile in the ordered dataset.
- \( P \) is the desired percentile (e.g., 25, 50, 75).
- \( n \) is the number of observations in the dataset.

If \( L \) is an integer, the \( P \)-th percentile is the value at the \( L \)-th position in the ordered dataset. If \( L \) is not an integer, we interpolate between the two nearest values.

#### Step 3: Apply the Formula to Each Percentile

##### Dataset:
$$
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
$$
Here, \( n = 10 \).

##### 1. Calculate the 25th Percentile (\( P = 25 \))
$$
L = \frac{25}{100} \times (10 + 1) = 0.25 \times 11 = 2.75
$$

Since \( L = 2.75 \) is not an integer, we interpolate between the 2nd and 3rd values in the dataset:
- The 2nd value is \( 2 \).
- The 3rd value is \( 3 \).

The interpolation formula is:
$$
\text{Percentile Value} = \text{Lower Value} + (\text{Fractional Part}) \times (\text{Higher Value} - \text{Lower Value})
$$

Here:
- Lower Value = 2
- Higher Value = 3
- Fractional Part = 0.75

$$
\text{25th Percentile} = 2 + 0.75 \times (3 - 2) = 2 + 0.75 = 2.75
$$

##### 2. Calculate the 50th Percentile (\( P = 50 \))
$$
L = \frac{50}{100} \times (10 + 1) = 0.5 \times 11 = 5.5
$$

Since \( L = 5.5 \) is not an integer, we interpolate between the 5th and 6th values in the dataset:
- The 5th value is \( 5 \).
- The 6th value is \( 6 \).

Using the interpolation formula:
$$
\text{Percentile Value} = \text{Lower Value} + (\text{Fractional Part}) \times (\text{Higher Value} - \text{Lower Value})
$$

Here:
- Lower Value = 5
- Higher Value = 6
- Fractional Part = 0.5

$$
\text{50th Percentile} = 5 + 0.5 \times (6 - 5) = 5 + 0.5 = 5.5
$$

##### 3. Calculate the 75th Percentile (\( P = 75 \))
$$
L = \frac{75}{100} \times (10 + 1) = 0.75 \times 11 = 8.25
$$

Since \( L = 8.25 \) is not an integer, we interpolate between the 8th and 9th values in the dataset:
- The 8th value is \( 8 \).
- The 9th value is \( 9 \).

Using the interpolation formula:
$$
\text{Percentile Value} = \text{Lower Value} + (\text{Fractional Part}) \times (\text{Higher Value} - \text{Lower Value})
$$

Here:
- Lower Value = 8
- Higher Value = 9
- Fractional Part = 0.25

$$
\text{75th Percentile} = 8 + 0.25 \times (9 - 8) = 8 + 0.25 = 8.25
$$

Final Answers:

$$
\boxed{2.75, 5.5, 8.25}
$$