Quartiles Formula

Quartiles Formula

Quartiles are a useful statistical tool in business statistics for dividing a dataset into four equal parts, each representing 25% of the data. They are often used in conjunction with other measures of central tendency and variability, such as the mean and standard deviation, to better understand the distribution of the data.

The quartiles of a dataset can be calculated using the following formula:

Q1 = L + (N/4 - F) * (U - L)/C

Q2 = L + (N/2 - F) * (U - L)/C

Q3 = L + (3N/4 - F) * (U - L)/C

where:Q1, Q2, and Q3 represent the first quartile, second quartile (which is equivalent to the median), and third quartile, respectively.

L represents the lower limit of the first class interval.

U represents the upper limit of the last class interval.

N represents the total number of observations in the dataset.

F represents the cumulative frequency of the class interval that contains the first quartile, second quartile, or third quartile.

C represents the width of each class interval.

To better understand how the quartiles formula works, let's consider an example. Suppose we have the following dataset of 20 observations:

{3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26}

To calculate the quartiles for this dataset, we first need to determine the class intervals. In this example, we could use class intervals of 3-7, 8-12, 13-17, and 18-26, with a class width of 5. We can then use a frequency table to determine the frequency of observations in each class interval:

Class Interval	Frequency
03-07	2
08-12	6
13-17	7
18-26	5

To calculate the first quartile (Q1), we need to find the cumulative frequency that corresponds to the 25th percentile of the data. We can do this by multiplying the total number of observations (N) by 0.25, which gives us 5. We then look for the class interval that contains the 5th observation (i.e., the observation that falls at the 25th percentile), which is the 8-12 class interval. The lower limit of this class interval is 8, the upper limit is 12, and the cumulative frequency for this class interval is 2. Therefore, we can use the quartiles formula to calculate Q1 as follows:

Q1 = 8 + (5 - 2) * 5/6 = 9.17

To calculate the second quartile (Q2), we simply need to find the median of the dataset, which is the value that falls at the 50th percentile. In this example, there are an even number of observations, so the median is the average of the two middle values: (15 + 16)/2 = 15.5.

To calculate the third quartile (Q3), we need to find the cumulative frequency that corresponds to the 75th percentile of the data. We can do this by multiplying the total number of observations (N) by 0.75, which gives us 15. We then look for the class interval that contains the 15th observation (i.e., the observation that falls at the 75th percentile), which is the 13-17 class interval. The lower limit of this class interval is 13, the upper limit is 17, and the cumulative frequency for this class interval is 9 (i.e., the sum of the frequencies for the 8-12 and 13-17 class intervals). Therefore, we can use the quartiles formula to calculate Q3 as follows:

Q3 = 13 + (15 - 9) * 5/7 = 17.14

Now that we have calculated the quartiles for our dataset, we can use them to gain insight into the distribution of the data. For example, we can use the interquartile range (IQR), which is the difference between the third and first quartiles (Q3 - Q1), to measure the spread of the middle 50% of the data. In our example, the IQR is 7.97 (17.14 - 9.17), which indicates that the middle 50% of the data is relatively tightly clustered around the median.

We can also use quartiles to identify outliers in the data. Any observation that falls more than 1.5 times the IQR above the third quartile or below the first quartile is considered an outlier. In our example, the upper and lower limits for outliers are 29.75 (17.14 + 1.5 * 7.97) and 0.56 (9.17 - 1.5 * 7.97), respectively. Since there are no observations outside of these limits in our dataset, we can conclude that there are no outliers.

In summary, quartiles are a powerful tool for summarizing and understanding the distribution of a dataset. By dividing the data into four equal parts, we can gain insight into the spread and shape of the data, as well as identify potential outliers. While the quartiles formula can seem daunting at first, with a little practice it becomes a simple and powerful tool for any data analyst. We also can check on limitations of statistics in the quatiles to particluar quadrant.