How to Calculate Quartiles for Grouped Data


Quartiles are values that split up a dataset into four equal parts.

You can use the following formula to calculate quartiles for grouped data:

Qi = L + (C/F) * (iN/4 – M)

where:

  • L: The lower bound of the interval that contains the ith quartile
  • C: The class width
  • F: The frequency of the interval that contains the ith quartile
  • N: The total frequency
  • M: The cumulative frequency leading up to the interval that contains the ith quartile

The following example shows how to use this formula in practice.

Example: Calculate Quartiles for Grouped Data

Suppose we have the following frequency distribution:

Now suppose we’d like to calculate the value at the third quartile (Q3) of this distribution.

The value at the third quartile will be located at position (iN/4) in the distribution.

Thus, (iN/4) = (3*92/4) = 69.

The interval that contains the third quartile will be the 21-25 interval since 69 is between the cumulative frequencies of 58 and 70.

Knowing this, we can find each of the values necessary to plug into our formula:

L: The lower bound of the interval that contains the ith quartile

  • The lower bound of the interval is 21.

C: The class width

  • The class width is calculated as 25 – 21 = 4.

F: The frequency of the interval that contains the ith quartile

  • The frequency of the 21-25 class is 12

N: The total frequency

  • The total cumulative frequency in the table is 92.

M: The cumulative frequency leading up to the interval that contains the ith quartile

  • The cumulative frequency leading up to the 21-25 class is 58.

We can then plug in all of these values into the formula from earlier to find the value at the third quartile:

  • Qi = L + (C/F) * (iN/4 – M)
  • Q3 = 21 + (4/12) * ((3)(92)/4 – 58)
  • Q3 = 24.67

The value at the third quartile is 24.67.

You can use a similar approach to calculate the values for the first and second quartiles.

Additional Resources

The following tutorials provide additional information for working with grouped data:

How to Find Mean & Standard Deviation of Grouped Data
How to Find the Mode of Grouped Data
How to Find the Median of Grouped Data
Grouped vs. Ungrouped Frequency Distributions

7 Replies to “How to Calculate Quartiles for Grouped Data”

    1. Hi Gao…Determining whether data is discrete or continuous is fundamental in statistics and data analysis. Here’s how you can differentiate between the two:

      ### Definitions

      – **Discrete Data**: This type of data consists of distinct, separate values. Discrete data often involves counts of items or events and is typically represented by integers. Examples include the number of students in a class, the number of cars in a parking lot, or the number of heads in coin flips.

      – **Continuous Data**: This type of data can take any value within a given range and is often represented by real numbers. Continuous data usually involves measurements and can be infinitely divisible. Examples include height, weight, temperature, and time.

      ### Steps to Determine if Data is Discrete or Continuous

      1. **Understand the Nature of the Data**:
      – **Countable**: If the data can only take specific values and there are gaps between these values, it is likely discrete. For example, the number of children in a family.
      – **Measurable**: If the data can take any value within a range and can be measured to any desired level of precision, it is likely continuous. For example, the height of students in a class.

      2. **Examine the Data Types**:
      – **Integer Values**: Data that are integers (whole numbers) are typically discrete. Examples include the number of books on a shelf, the number of students attending a class, etc.
      – **Real Numbers**: Data that include decimal points and fractions are usually continuous. Examples include the temperature of a city, the length of an object, etc.

      3. **Check the Context and Units**:
      – **Categories and Counts**: If the data represent categories or counts of items/events, it is discrete. For example, the number of defective items in a batch.
      – **Measurements and Intervals**: If the data represent measurements that can take any value within an interval, it is continuous. For example, the time taken to run a marathon.

      ### Examples

      #### Discrete Data Examples
      – Number of students in a classroom.
      – Number of cars in a parking lot.
      – Number of questions answered correctly on a test.

      #### Continuous Data Examples
      – The height of individuals.
      – The weight of fruits in a grocery store.
      – The temperature readings throughout a day.

      ### Practical Approach

      Here are some practical steps to help you determine if your data is discrete or continuous:

      #### Step 1: Inspect the Data
      Look at a sample of your data. If you see values like 1, 2, 3, etc., and no fractions or decimals, it is likely discrete. If you see values like 1.5, 2.7, etc., it is likely continuous.

      #### Step 2: Analyze the Data Source
      Consider how the data was collected:
      – If it was counted (e.g., number of occurrences), it is likely discrete.
      – If it was measured (e.g., weight, height), it is likely continuous.

      #### Step 3: Use Statistical Tests
      While not commonly needed, you can use statistical methods to confirm the nature of the data:
      – **Histogram**: Plot a histogram of your data. Discrete data will show distinct bars, while continuous data will show a smoother distribution.
      – **Kolmogorov-Smirnov Test**: For more advanced analysis, use tests like the Kolmogorov-Smirnov test to check for continuous distributions.

      ### Conclusion

      To determine if data is discrete or continuous, consider the nature of the data, the types of values it takes, the context of data collection, and how it is measured or counted. Discrete data involves distinct, countable values, while continuous data involves measurable quantities that can take any value within a range. Understanding these characteristics will help you categorize your data correctly and choose appropriate statistical methods for analysis.

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