The Median and Quartiles

Quartiles

Quartiles are values that divide a dataset into four equal parts, each containing 25% of the data. The three quartiles are denoted as Q1, Q2 (which is the median), and Q3.

Key Facts

  1. Quartiles: Quartiles are values that divide a dataset into four equal parts, each containing 25% of the data. The three quartiles are denoted as Q1, Q2 (which is the median), and Q3.
  2. Median: The median is the middle value of a dataset when it is arranged in ascending or descending order. It divides the dataset into two equal halves.
  3. Finding the Median of a Quartile: To find the median of a quartile, you need to calculate the quartiles first. Here’s how you can do it:

    a. Order your dataset from lowest to highest values.

    b. Find the median (Q2) of the entire dataset. This will be the second quartile.

    c. Split the dataset into two halves at the median (Q2). The lower half will contain values less than Q2, and the upper half will contain values greater than Q2.

    d. The median of the lower half, excluding Q2, will be the first quartile (Q1).

    e. The median of the upper half, excluding Q2, will be the third quartile (Q3).

  4. Interquartile Range (IQR): The interquartile range is the range between the first quartile (Q1) and the third quartile (Q3). It represents the spread of the middle 50% of the data.

Median

The median is the middle value of a dataset when it is arranged in ascending or descending order. It divides the dataset into two equal halves.

Finding the Median of a Quartile

To find the median of a quartile, you need to calculate the quartiles first. Here’s how you can do it:

  1. Order your dataset from lowest to highest values.
  2. Find the median (Q2) of the entire dataset. This will be the second quartile.
  3. Split the dataset into two halves at the median (Q2). The lower half will contain values less than Q2, and the upper half will contain values greater than Q2.
  4. The median of the lower half, excluding Q2, will be the first quartile (Q1).
  5. The median of the upper half, excluding Q2, will be the third quartile (Q3).

Interquartile Range (IQR)

The interquartile range is the range between the first quartile (Q1) and the third quartile (Q3). It represents the spread of the middle 50% of the data.

Example

Let’s consider the following dataset:

23, 32, 33, 47, 40, 43, 44, 47, 52
  1. Order the dataset23, 32, 33, 40, 43, 44, 47, 47, 52
  2. Find the median (Q2)The median is 43.
  3. Split the dataset into two halvesThe lower half is 23, 32, 33, 40, and the upper half is 44, 47, 47, 52.
  4. Find the first quartile (Q1)The median of the lower half, excluding 43, is 32.5.
  5. Find the third quartile (Q3)The median of the upper half, excluding 43, is 47.

Therefore, the quartiles of the given dataset are Q1 = 32.5, Q2 = 43, and Q3 = 47. The interquartile range is IQR = Q3 – Q1 = 47 – 32.5 = 14.5.

Sources

FAQs

What is a quartile?

A quartile is a value that divides a dataset into four equal parts, each containing 25% of the data. The three quartiles are denoted as Q1, Q2 (which is the median), and

What is the median?

The median is the middle value of a dataset when it is arranged in ascending or descending order. It divides the dataset into two equal halves.

How do I find the median of a quartile?

To find the median of a quartile, you need to calculate the quartiles first. Here’s how you can do it:

  1. Order your dataset from lowest to highest values.
  2. Find the median (Q2) of the entire dataset. This will be the second quartile.
  3. Split the dataset into two halves at the median (Q2). The lower half will contain values less than Q2, and the upper half will contain values greater than
  4. The median of the lower half, excluding Q2, will be the first quartile (Q1).
  5. The median of the upper half, excluding Q2, will be the third quartile (Q3).

What is the interquartile range (IQR)?

The interquartile range is the range between the first quartile (Q1) and the third quartile (Q3). It represents the spread of the middle 50% of the data.

How do I interpret the quartiles and IQR?

The quartiles and IQR can be used to understand the distribution of a dataset. The median (Q2) represents the middle value, while the quartiles (Q1 and Q3) represent the upper and lower bounds of the middle 50% of the data. The IQR represents the spread of the middle 50% of the data.

What is the difference between the median and the mean?

The median is the middle value of a dataset, while the mean is the average of all the values in a dataset. The median is not affected by outliers, while the mean can be affected by outliers.

When should I use the median instead of the mean?

The median should be used instead of the mean when the data is skewed or has outliers. The median is a more robust measure of central tendency than the mean.

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