**Calculation of Quartiles, Deciles & Percentiles from Grouped Data**

• Quartiles

*ℓ*= Lower boundary of the quartile class

*f*= Frequency of quartile class

‹

*C*= Cumulative frequency of the preceding quartile class

*w*= length of the quartile class

• Deciles

*ℓ*= Lower boundary of the decile class

*f*= Frequency of decile class

‹

*C*= Cumulative frequency of the preceding decile class

*w*= length of the decile class

• Percentiles

*ℓ*= Lower boundary of the percentile class

*f*= Frequency of percentile class

‹

*C*= Cumulative frequency of the preceding percentile class

*w*= length of the percentile class

Example 1: The following table shows the Price of 80 New Vehicles Sold Last Month at Toyota (in $ thousand) is given below.

Selling Price 15-17 18-20 21-23 24-26 27-29 30-32 33-35 Total

Freq 8 23 17 18 8 4 2 80

Calculate 3rd quartile, 3rd decile, 68th percentile for the previously given data.

Solution:

Selling Price | Frequency | Cumulative Frequency |
---|---|---|

15-17 | 8 | 8 |

18-20 | 23 | 31 |

21-23 | 17 | 48 |

24-26 | 18 | 66 |

27-29 | 8 | 74 |

30-32 | 4 | 78 |

33-34 | 2 | 80 |

Total | 80 |

• 3rd Quartile

Here ¾*N*=60. So 3rd quartile lies in 24-26 class

Lower boundary of the class where median lies is (

*ℓ*).

• 3rd Decile

Here 3

*N*/10= 24. So 3rd quartile lies in 18-20 class.

Lower boundary of the class where median lies is (

*ℓ*).

• 68th Percentile

Here 68

*N*/100= 54.4. So 3rd quartile lies in 24-26 class.

Example 2:

For the following distribution find:

(i) the median.

(ii) the first quartile.

(iii) the third quartile.

(iv) the 10th percentile.

(v) the 6th decile.

Class | f |
---|---|

3-6 6-9 9-12 12-15 15-18 |
4 10 12 9 5 |

Solution:

Class boundary | f |
c.f |
---|---|---|

3-6 6-9 9-12 12-15 15-18 |
4 10 12 9 5 |
4 14 26 35 40 |

N=40 |

(i) median =½*N*th observation

=20th observation, which lies in the class 9—12.

⇒ median class = 9—12. Therefore,

(ii)

*Q*

_{1}=¼

*N*th observation

=10th observation, which lies in the class 6—9.

⇒

*Q*

_{1}c1ass = 6—9. Therefore,

(iii)

*Q*

_{3}=¾

*N*th observation

=30th observation, which lies in the Class 12—15.

⇒

*Q*

_{3}class = 12—15. Therefore,

(iv)

*P*

_{10}=10

*N*/100th observation

= 4th observation, which lies in the class 3—6.

⇒

*P*

_{10}c1ass = 3—6. Therefore,

(v)

*D*

_{6}=6

*N*/10th observation

= 24th observation, which lies in the Class 9—12.

⇒

*D*

_{6}C1ass = 9—12. Therefore,

Example 3:

The following is the frequency distribution of 30 automobiles tested for fuel efiiciency.

Kilometer/litre | Frequency |
---|---|

8-12 13-17 18-22 23-27 28-32 |
3 5 15 5 2 |

Construct a cumulative percentile graph and hence find:

(a) the median.

(b) the percentile rank of 17.5.

(c) the lower quartile.

(d) the 7th decile.

(e) the 10th percentile.

Solution:

(a) median:10 kilometer/litre.

(b) percentile rank of 17.5=28%.

(c)

*Q*

_{1}=16.6 kilometer/litre.

(d)

*D*

_{7}=22 kilometer/litre.

(e)

*P*

_{10}=12 kilometer/litre.

Percentiles:

Percentiles are the values of the variable that divide a set of observations into 100 equal parts. Each set of observations has 99 percentiles and are denoted by *P*_{1}, *P*_{2}, . . . , *P*_{99}.

The kth percentile, *P _{k}* is a value such that

*k*% of the observations are smaller than or equal to

*P*and (100-

_{k}*k*)% of the observations are larger than

*P*. For example, if a value is located at the 80th percentile, it means that 80% of the values that fall below the value and 20% of the values fall above it.

_{k}Notes:

(i) A percentile is a value in the data set.

(ii) A percentile rank of a given value is a percent that indicates the percentage of data is smaller than the value.

Deciles:

Deciles are the values of the variable that divide a set of observations into 10 equal parts. Each set of observations has 9 deciles are denoted by *D*_{1}, *D*_{2}, …, *D*_{9}.

The first decile *D*_{1} is a value in the data set that 10% of the values fall below *D*_{1} and 90% of the values fall above *D*_{1}.

Similarly, the second decile *D*_{2} is a value in the data set that 20% of the values fall below *D*_{2} and 80% of the values fall above *D*_{2} and so on.

Note: The first decile and tenth percentile are the same i.e. *D*_{1}=*P*_{10}.

Sirnilarly, *D*_{2}=*P*_{20}, …, *D*_{3}=*P*_{30}, *D*_{9}=*P*_{90}.

Quartiles:

Quartiles are the values of the variable that divide a set of observations into 4 equal parts. Each set of observations has 3 quartiles and they are denoted by *Q*_{1}, *Q*_{2} and *Q*_{3}.

The first quartile *Q*_{1} is a value in the data set that 25% of the values fall below *Q*_{1} and 75% of the values fall above *Q*_{1}.

The second quartile *Q*_{2} is a value in the data set that 50% of the values fall below *Q*_{2} and 50% of the values fall above *Q*_{2}.

The third quartile *Q*_{3} is a value in the data set that 75% of the values fall below *Q*_{3} and 25% of the values fall above *Q*_{3}.

Note: *Q*_{1}=*P*_{25}, *Q*_{2}=*P*_{50}, *Q*_{3}=*P*_{75}.

Median: The 50th percentile, 5th decile and second quartile of a distribution are equal to the same value and are referred to as the median. That is

median=*Q*_{2}=*D*_{5}=*P*_{50}.

Percentile graph:

The percentile graph is similar to ogive. To construct a percentile graph the upper bound- aries are plotted on the x—axis and cumulative percentage are plotted on the y—axis. Using a percentile graph one can find the approximate percentile rank for a given value and find the approximate value of data for a given percentile rank.

Let’s read post Drawing Box-and-Whisker Plots (Boxplots).