Percentiles & Quartiles: Videos & Practice Problems

Percentiles indicate the percentage of data points below a specific value in a dataset. To calculate a percentile, use the formula: $\frac{n}{N}100$, where n is the number of responses below the value and N is the total number of responses. Special percentiles include quartiles, with Q1 as the 25th percentile and Q3 as the 75th percentile. The interquartile range (IQR) is the difference between Q3 and Q1, indicating data variability.

Percentiles and Quartiles

In statistics, understanding how a specific value relates to a dataset is crucial, particularly when determining the percentage of responses that fall below that value. This is where the concept of percentiles becomes essential. A percentile indicates the percentage of data points in a dataset that are less than a given value. For instance, if a test score is in the eightieth percentile, it means that the score is higher than 80% of the other scores in the dataset.

To calculate the percentile of a specific value, you can use the formula:

P = (N_b / N_t) × 100

where P is the percentile, N_b is the number of responses below the value, and N_t is the total number of responses in the dataset. For example, if you want to find the percentile of a score of 1280 in a dataset of SAT scores, you would first count how many scores are below 1280 and divide that by the total number of scores, then multiply by 100. If there are 6 scores below 1280 in a dataset of 12, the calculation would yield a percentile of 50, indicating that a score of 1280 is at the fiftieth percentile.

Conversely, you can also determine the score corresponding to a specific percentile. For example, to find the seventy-fifth percentile (P₇₅), you would need to calculate how many values fall below that score. Using the formula, if you find that 9 values are below the score, you can then identify the score that corresponds to that position in the ordered dataset. In this case, if the ninth score is 1360 and the tenth score is 1420, any score between these two would represent the seventy-fifth percentile. A common approach is to take the average of these two scores, resulting in a value of 1390 for P₇₅.

Special percentiles known as quartiles are also important. The first quartile (Q₁) represents the twenty-fifth percentile, while the third quartile (Q₃) corresponds to the seventy-fifth percentile. The second quartile (Q₂) is the median, or fiftieth percentile. To find Q₁ and Q₃, you can apply the same percentile formula. For instance, if you calculate Q₁ and find that it lies between the third and fourth scores in a dataset, you would average those two scores to determine its value.

Another important concept is the interquartile range (IQR), which measures the spread of the middle fifty percent of the data. It is calculated as:

IQR = Q₃ - Q₁

For example, if Q₃ is 1390 and Q₁ is 1195, the IQR would be 195, indicating the range within which the central half of the data lies.

Understanding these concepts of percentiles, quartiles, and the interquartile range is fundamental for analyzing and interpreting data effectively.

Answer the questions below using the data in the table.

(A) Find $P_{50}$
(B) A playlist with 15 songs is in which percentile?
(C) Find Q1 and Q3

• A. (A) $16$; (B) $43^{rd}$ percentile; (C) $Q1=7$ & $Q3=21$
• B. (A) $16$; (B) ${12}^{\mathrm{th}}$ percentile; (C) $Q1=12$ & $Q3=20.5$
• C. (A) $15.5$; (B) ${43}^{rd}$ percentile; (C) $Q1=12$ & $Q3=20.5$
• D. (A) $15.5$; (B) ${12}^{\mathrm{th}}$ percentile; (C) $Q1=7$ & $Q3=21$

Answer: (C)

Percentiles and Quartiles Example 1

To find the first quartile (Q1) and the third quartile (Q3) of a dataset, we utilize the concept of percentiles. Q1 represents the 25th percentile, while Q3 corresponds to the 75th percentile. The process begins with organizing the dataset in numerical order, which simplifies the identification of values below certain thresholds.

The percentile formula is expressed as:

P = \frac{N_b}{N} \times 100

where P is the percentile, N_b is the number of values below the desired percentile value, and N is the total number of responses in the dataset.

For Q1, we set P to 25:

25 = \frac{N_b}{10} \times 100

Solving for N_b, we find:

N_b = \frac{25 \times 10}{100} = 2.5

Since N_b is a decimal, we round down to the nearest whole number, which is 2. This indicates that we should select the value in the dataset with two values below it. Thus, Q1 is the third value in the ordered dataset, which is 24.

Next, we calculate Q3 by setting P to 75:

75 = \frac{N_b}{10} \times 100

Again, solving for N_b gives us:

N_b = \frac{75 \times 10}{100} = 7.5

Rounding down, we take the whole number part, which is 7. This means we select the value in the dataset with seven values below it. Therefore, Q3 is the eighth value in the ordered dataset, which is 42.

In summary, for the given dataset, Q1 is 24 and Q3 is 42, providing valuable insights into the distribution of the data.

Here’s what students ask on this topic:

What is a percentile, and how is it calculated?

A percentile indicates the percentage of data points in a dataset that fall below a specific value. To calculate a percentile, use the formula:

$\frac{n}{N}100$

Here, $n$ is the number of responses below the value of interest, and $N$ is the total number of responses in the dataset. Multiply the result by 100 to express it as a percentage. For example, if 6 out of 12 responses are below a value, the percentile is:

$\frac{6}{12}100=50$

This means the value is at the 50th percentile.

What are quartiles, and how are they related to percentiles?

Quartiles divide a dataset into four equal parts, each containing 25% of the data. They are specific percentiles:

• Q1 (First Quartile): The 25th percentile, where 25% of the data falls below this value.
• Q2 (Second Quartile): The 50th percentile, also known as the median, where 50% of the data falls below this value.
• Q3 (Third Quartile): The 75th percentile, where 75% of the data falls below this value.

Quartiles help summarize the distribution of data and are used to calculate the interquartile range (IQR), which measures data variability. The IQR is the difference between Q3 and Q1:

$\mathrm{IQR}=Q_3-Q_1$

How do you find the 75th percentile in a dataset?

To find the 75th percentile (P₇₅) in a dataset, follow these steps:

1. Order the dataset from smallest to largest.
2. Calculate the position of the 75th percentile using the formula:

$\mathrm{Position}=\frac{75}{100}N$

Here, $N$ is the total number of data points.

3. If the position is an integer, the value at that position is the 75th percentile. If not, take the average of the two closest values.

For example, in a dataset of 12 values, the position is:

$\frac{75}{100}12=9$

The 9th value (or average of the 9th and 10th values) represents the 75th percentile.

What is the interquartile range (IQR), and how is it calculated?

The interquartile range (IQR) measures the spread of the middle 50% of a dataset. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1):

$\mathrm{IQR}=Q_3-Q_1$

Q1 is the 25th percentile, and Q3 is the 75th percentile. For example, if Q3 = 1390 and Q1 = 1195, the IQR is:

$1390-1195=195$

The IQR is useful for identifying variability and detecting outliers in a dataset.

How do you interpret a test score in the 80th percentile?

A test score in the 80th percentile means that the score is higher than 80% of the other scores in the dataset. For example, if 100 students took a test and your score is in the 80th percentile, it indicates that your score is better than 80 students, while 20 students scored higher than you. Percentiles provide a relative measure of performance compared to others in the dataset, making them useful for understanding rankings and distributions.