Interquartile deviation is a very important statistical concept in data analysis. This term is often used in statistics to measure the level of variance or spread of data in a set of values.
This concept is also called the quasi-interquartile range or interquartile deviation. This semi-interquartile range allows you to understand how the data is distributed in more depth, especially when you have grouped or grouped data.
In this discussion, you will explain clearly and simply what spring deflection is, how to calculate it, and what is the interpretation of the spring deflection value.
What is the interquartile deviation?
Quartiles is a term used to divide sequential data into four parts containing the same amount of data in each part. In statistical analysis, there are three quartile values, namely the lower quartile (Q1), the middle quartile (Q2), and the upper quartile (Q3).
The interquartile range itself is the difference between the upper quartile value (Q3) and the lower quartile value (Q1). To calculate the quartile deviation, you need to determine the values of Q3 and Q1 first. The interquartile deviation is actually the average distance between the second quartile (Q2) and the first quartile (Q1) or third quartile (Q3).
What is upper quadrant deviation? The upper quartile deviation is calculated as half of the interquartile range measured from the upper quartile value (Q₃) to the highest value in the data.
Meanwhile, the lower interquartile range is calculated as half of the interquartile range measured from the lowest value in the data to the lower quartile value (Q₁).
Apart from this, there is another statistical measure known as standard deviation. Standard deviation is a deviation used to describe the spread of data around the mean value.
The way to calculate standard deviation is to measure how far each data point deviates from the average value, and then take the average of these distances.
These three measures, which are the upper half of the interquartile range, the lower interquartile deviation, and the standard deviation, have different roles in analyzing the data and providing an overview of the distribution of the data and the level of variation present in the data.
Using these metrics helps provide a more comprehensive understanding of the characteristics of the data and facilitates drawing conclusions in statistical analysis.
What is the function of the interquartile deviation?
The interquartile range has various functions in data analysis, including:
1. Measure data spread
The semi-interquartile range is used to evaluate the spread of data across quartile values. The greater the interquartile deviation, the greater the spread of the data.
2. Detect outliers
Outliers are data that are far from the median or interquartile value. The interquartile range plays an important role in determining the presence of outliers in the data. If there is data that is far outside the interquartile range, the data can be considered outliers.
3. Compare the distribution of data between groups
Interquartile deviation is used to compare the spread of data between groups. If the quartile deviation in group A is larger than in group B, it can be concluded that the data in group A is more spread out than in group B.
4. Determine the limits of the normal value
Next, the quartile deviation function helps determine the limits of the normal value of the data. The normal value limit can be calculated by taking the lower quartile value (Q₁) minus 1.5 times the spring deflection, and the upper quartile value (Q₃) plus 1.5 times the spring deflection.
Data that falls outside the normal value limits can be considered abnormal data or outliers.
By understanding the different quartile deviation functions, you can apply them in data analysis to gain more in-depth information about the distribution and characteristics of your data.
Interquartile deviation formula
Before continuing to understand formulas, it is necessary to understand the difference between individual data and group data first. Individual data is data that is simply presented, does not contain time periods, and is not very large in quantity.
Meanwhile, group data is data that is collected in the form of a time interval. For example, data can be grouped into the ranges 1 to 5, 6 to 10, and so on. The amount of data in this type is larger and is often displayed in a frequency table.
1. Single data
The quartile deviation formula for individual data and group data has differences in how the lower quartile (Q₁) and upper quartile (Q₃) values are calculated. Below are the differences in the quartile deviation formula for individual data and group data.
First, the data is sorted sequentially. The lower quartile value (Q₁) is obtained from the data value at position n/4, and the upper quartile value (Q₃) is obtained from the data value at position 3n/4, where n is the amount of data.
Then, the quartile deviation can be calculated using the formula quartile deviation (Qd) = ½ (Q₃ – Q₁)
2. Group data
The first step is to determine the frequency class of the group data. The lower quartile value (Q₁) is obtained from the data value at the boundary of the lower layer of the class where the median is located, and the upper quartile value (Q₃) is obtained from the data value at the boundary of the upper layer of the class where the median is located. Then, the spring deflection can be calculated using the formula Qd = ½ (Q₃ – Q₁)
So, the difference in the interquartile deviation formula for individual data and group data lies in how the lower quartile (Q₁) and upper quartile (Q₃) values are obtained. Although the formula is the same, the way to calculate lower quartile and upper quartile values differs, depending on the type of data you have.
Example of interquartile deviation questions
Below are some examples of questions that can increase your understanding of interquartile deviation.
Question No. 1
Question: Given the following height data for high school students: 160, 165, 170, 155, 175, 162, 168, 160, 158, 172. Calculate the interquartile deviation from these data.
Discussion: The first step is to sort the data from smallest to largest: 155, 158, 160, 160, 162, 165, 168, 170, 172, 175. Next, determine the lower quartile (Q₁) and upper quartile (Q₃).
In this case, since there are 10 statements, Q₁ is in the third position (second index) and Q₃ is in the eighth position (seventh index). Q₁ = 160 Q₃ = 170 Next, calculate the quarter deviation using the formula: Qd = (Q₃ – Q₁) / 2 = (170 – 160) / 2 = 5
Therefore, the interquartile deviation of high school students’ height data is 5.
Question 2
Question: Below is a spreadsheet of the number of product sales in thousands of rupees at store A for 10 days:
| day | Number of sales |
| 1 | 50 |
| 2 | 60 |
| 3 | 55 |
| 4 | 70 |
| 5 | 65 |
| 6 | 75 |
| 7 | 80 |
| 8 | 85 |
| 9 | 90 |
| 10 | 95 |
Calculate the interquartile deviation of sales data.
Discussion: The first step is to sort the data from smallest to largest: 50, 55, 60, 65, 70, 75, 80, 85, 90, 95. Then determine the lower quartile (Q₁) and upper quartile (Q₃).
In this case, since there are 10 statements, Q₁ is in the third position (second index) and Q₃ is in the eighth position (seventh index). Q₁ = 60 Q₃ = 85 Next, calculate the quarter deviation using the formula: Qd = (Q₃ – Q₁) / 2 = (85 – 60) / 2 = 12.5
So, the interquartile deviation of product sales data at Store A is 12.5.
Question 3
Problem: A bookstore records book sales for 10 consecutive days. Here are the book sales data in thousands of rupees: 50, 60, 55, 70, 65, 75, 80, 85, 90, 95. Calculate the interquartile deviation of the sales data.
Discussion: The first step is to sort the data in order from smallest to largest: 50, 55, 60, 65, 70, 75, 80, 85, 90, 95. Next, determine the lower quartile (Q₁) and upper quartile (Q₃).
In this case, since there are 10 statements, Q₁ is in the third position (second index) and Q₃ is in the eighth position (seventh index). x₁ = 60 x₃ = 85
Next, use the formula Quartile Deviation (Qd) = ½ (Q₃ – Q₁) = ½ (85 – 60) = 12. So, the quartile deviation of the 10-day book sales data is Rs 12.5 lakh.
Question 4
Question: The company is conducting a selection test for potential employees. Choice test score data are grouped into several frequency categories with a specific value range. The data is as follows:
| Season | Frequency value range |
| 1 | 50 – 59 |
| 2 | 60 – 69 |
| 3 | 70 – 79 |
| 4 | 80 – 89 |
| 5 | 90 – 100 |
The company wants to calculate the interquartile deviation from the choice test score data. Calculate the quarterly deviation from the table.
Discussion: First of all, it is necessary to calculate the frequency of each category. Each category has its own frequency, for example: Category 1: 4 Category 2: 6 Category 3: 8 Category 4: 10 Category 5: 12. Next, determine the average value for each category.
In this context, the average value for each category can be calculated by summing the lower and upper limits of the category’s value range, and then dividing the result by two. For example: Category 1: (50 + 59) / 2 = 54.5 Category 2: (60 + 69) / 2 = 64.5 Category 3: (70 + 79) / 2 = 74.5 Category 4: (80 + 89) / 2 = 84.5 Score 5: (90 + 100) / 2 = 95
Next, calculate the total frequency for all categories. The total frequency in this example is 40. Determine the lower quartile (Q₁) and the upper quartile (Q₃).
In this case, since there are 40 data, Q₁ is at the 10th position (9th index) and Q₃ is at the 30th position (29th index). Q₁ = 64.5 Q₃ = 90. Finally, calculate the quarter deviation using the formula: Qd = (Q₃ – Q₁) / 2 = (90 – 64.5) / 2 = 12.75
Therefore, the interquartile deviation of the prospective employee selection test result data is 12.75.
Close
These examples of skew quarter questions can help you understand the material well. You can continue practicing until your understanding improves. Discussing the different questions is sure to help you complete the tests related to this subject.