The 40th percentile is a statistical measure that indicates the value below which 40% of the data points in a dataset fall. In other words, if you have a set of values arranged in ascending order, the 40th percentile is the value at which 40% of the data is less than or equal to that value, and 60% of the data is greater than that value.

### Detailed Explanation

**Understanding Percentiles**: Percentiles divide a dataset into 100 equal parts. Each percentile represents a point in the data that reflects how the data points are distributed. While the 50th percentile represents the median (the middle value), other percentiles help in understanding the spread of the data.**Calculation of the 40th Percentile**:**Arrange Data**: Start by sorting the data in ascending order.**Locate the Rank**: To find the rank (position) corresponding to the 40th percentile, you can use the formula:

[

P = \frac{n \times k}{100}

]

where:- ( P ) is the rank,
- ( n ) is the total number of data points,
- ( k ) is the desired percentile (in this case, 40).

**Interpret the Rank**:- If ( P ) is a whole number, the value at that position is the 40th percentile.
- If ( P ) is not a whole number, round it up to the next whole number and take the average of the values at both the rounded value and the preceding position.

### Example

Consider the following dataset of exam scores:

[ 56, 72, 69, 81, 74, 88, 90, 65, 78 ]

**Sort the Data**:

[ 56, 65, 69, 72, 74, 78, 81, 88, 90 ]**Total Number of Data Points**:

[ n = 9 ]**Calculate the Position**:

[

P = \frac{9 \times 40}{100} = 3.6

]

Since 3.6 is not a whole number, round it up to 4.**Value at the 4th Position**(in the sorted list):

The 4th value is 72.

So, in this dataset, the 40th percentile is 72. This means that 40% of the scores are less than or equal to 72.

### Applications of Percentiles

**Education**: Percentiles are often used in education to interpret standardized test scores.**Health**: In health metrics, percentiles can show how a particular measurement compares to a reference population, such as growth charts for children.**Statistics**: Analysts use percentiles to summarize data points for reporting and analysis.

### Conclusion

The 40th percentile, thus, serves as a useful benchmark for understanding where a particular value stands within a dataset, providing insight into the distribution and relative standing of data points.