四分位数公式

四分位数概念用于将数据分为四个部分。它与中位数相同，它将给定数据分成两个相等的部分。这个四分位数的概念属于统计学的主题，统计学是对数据收集的研究，分析它，解释，呈现有组织的数据。

四分位数公式

如上所述，Quartile 将数据分成 4 个相等的部分。这可以通过下图直观地表示。

• 四分位数 1 位于开始学期和中期学期之间。
• 四分位数 2 位于开始学期和最后学期之间，即中期。
• 四分位数 3 位于四分位数 2 和最后一个学期之间。

连续变量与自变量

有一个单独的公式可以找到每个四分位数。为了首先找到这些四分位数，请将给定的数字序列数据按升序排序。

获取四分位数公式的步骤

1. 按升序对给定数据进行排序。
2. 根据需要从下面的公式中找到相应的四分位数/项。

First Quartile = ((n + 1)/4)^th term

Second Quartile = ((n + 1)/2)^th term

Third Quartile = (3(n + 1)/4)^th term

Where n is the total count of numbers in the given data.

So the generalized formula for the quartile is,

Quartile_r = l₁ + ((r(N/4) – cf)/f)×(l₂ – l₁)

Where,

Quartile_r indicates r^th quartile.

l₁, l₂ are lower limit and upperlimit value.

f is the frequency count.

cf is the cumulative frequency of class precedding the quartile class.

四分位距

四分位距是第一个四分位数和第三个四分位数之间的距离。它也被称为中间价差。它可以帮助我们计算分成四分位数的数据的变化。计算四分位距的公式由下式给出，

Interquartile range = Q₃ – Q₁

Where, Q₃ is third/upper quartile.

and Q₁ is first/lower quartile.

四分位偏差

四分位数偏差定义为第一个四分位数和第三个四分位数之间距离的一半。它也被称为半四分位数范围。四分位偏差的公式由下式给出，

Quartile Deviation = (Q₃ – Q₂)/2

让我们看几个例子来更好地理解这些概念，

问题 1：找到给定数据 10、30、5、12、20、40、25、15、18 的四分位数 1。

解决方案：

Step 1: Sort the given data in the ascending order

5, 10, 12, 15, 18, 20, 25, 30, 40

Step 2: Find 1^st Quartile

Quartile-1 = ((n + 1)/4)^th term

Here n = 9 because there are total 9 numbers in the given data.

First Quartile = ((9 + 1)/4)^th term

= (10/4)^th term

= 2.5^th term

2.5^th term is average result of 2nd and 3rd term = (10 + 12)/2 = 11

The First Quartile value is 11.

问题 2：找到数据 10、30、5、12、20、40、25、15、18 的第二个四分位数。

解决方案：

Step 1: Sort the given data in the ascending order

5, 10, 12, 15, 18, 20, 25, 30, 40

Step 2: Find 2^nd Quartile

Quartile-2 = ((n + 1)/2)^th term

Here n = 9 because there are total 9 numbers in the given data.

Second Quartile = ((9 + 1)/2)^th term

= (10/2)^th term

= 5^th term

5^th term is 18

So the second Quartile value is 18.

问题 3：找出数据 10、30、5、12、20、40、25、15、18 的第三个四分位数。

解决方案：

Step 1: Sort the given data in the ascending order

5, 10, 12, 15, 18, 20, 25, 30, 40

Step 2: Find 3^rd Quartile

Quartile-3 = (3(n + 1)/4)^th term

Here n = 9 because there are total 9 numbers in the given data.

Third Quartile = (3(9 + 1)/4)^th term

= (3(10)/4)^th term

= 7.5^th term

7.5^th term is average result of 7^th and 8^th term = (25 + 30)/2 = 27.5

So the third Quartile value is 27.5.

问题 4：找出数据 10、5、20、50、25 的第一个、第二个和第三个四分位数

解决方案：

Step 1: Sort the given data in the ascending order

5, 10, 20, 25, 50

Step 2: Find all Quartiles step by step

Quartile-1 = ((n + 1)/4)^th term

Here n = 5 because there are total 5 numbers in the given data.

First Quartile = ((5 + 1)/4)^th term

= (6/4)^th term

= 1.5^th term

1.5^th term is average result of 1^st and 2^nd term = (5 + 10)/2 = 7.5

The First Quartile value is 7.5.

Quartile-2 = ((n + 1)/2)^th term

Second Quartile = ((5 + 1)/2)^th term

= (6/2)^th term

= 3^rd term

3^rd term is 20

So the second Quartile value is 20.

Quartile-3 = (3(n + 1)/4)^th term

Third Quartile = (3(5 + 1)/4)^th term

= (3(6)/4)^th term

= 4.5^th term

4.5^th term is average result of 4^th and 5^th term = (25 + 50)/2 = 37.5

So the third Quartile value is 37.5.

问题 5：找出数据 1、10、9、7、5、11、20、19、17、15、12 的第三个四分位数。

解决方案：

Step 1: Sort the given data in the ascending order

1, 5, 7, 9, 10, 11, 12, 15, 17, 18, 20

Step 2: Find 3^rd Quartile

Quartile-3 = (3(n + 1)/4)^th term

Here n = 11 because there are total 11 numbers in the given data.

Third Quartile = (3(11 + 1)/4)^th term

= (3(12)/4)^th term

 = 9^th term

9^th term is 17

So the third Quartile value is 17.

问题 6：如果第一个四分位数是 10，第三个四分位数是 30 厘米，数据的四分位数范围是多少。

解决方案：

Given,

Q₁ = 10

Q₃ = 30

Interquartile range = Q₃ – Q₁

= 30 – 10

Interquartile range = 20

问题 7：如果第一个四分位数是 15，第三个四分位数是 30 厘米，那么数据的四分位数偏差是多少。

解决方案：

Given,

Q₁ = 15

Q₃ = 30

Quartile Deviation = (Q₃ – Q₁)/2

= (30 – 15)/2

= 15/2

Quartile Deviation = 7.5