Quartiles

Introduction to Quantiles and Quartiles

Quantiles are the techniques used to divide the data into different equal parts. For example, quantiles divide the data into four equal parts. Quartile comes from quarter which means 4th part. Deciles divide the data into ten equal parts and they come from deca means the 10th part. Percentiles divide the data into hundred parts and it comes to percent which means the 100th part.

Therefore, quartiles, deciles, and percentiles are used to divide the data into 4, 10, and 100 parts respectively. The quantiles, deciles, and percentiles are collectively called quantiles.

Quartiles

Quartiles are the rules that divide the data into four equal parts. When we divide any data into four equal parts, we cut it at equidistant points. The quartiles ($Q_1, Q_2$, and $Q_3$) divide the data into four equal parts, so divide the number of observations by four for each quartile.

Quartiles for Ungroup Data

\begin{align*}
Q_1 &= \left(\frac{n+1}{4}\right)th \text{ value is the} \frac{1}{4} \text{ part}\\
Q_2 &= \left(\frac{2(n+1)}{4}\right)th \text{ value is the} \frac{2}{4} \text{ part}\\
Q_3 &=\left(\frac{3(n+1)}{4}\right)th \text{ value is the} \frac{3}{4} \text{ part}
\end{align*}

The following ungroup data has 96 observations $(n=96)$

222225253030303131333639
404042424848505152555759
818689899091919192939393
939494949596969697979898
999999100100100101101102102102102
102103103104104104105106106106107108
108108109109109110111112112113113113
113114115116116117117117118118119121

The first, second, and third quartiles of the above data set are:

\begin{align*}
Q_1 &= \left(\frac{n}{4}\right)th \text{ position } = \left(\frac{96}{4} = 24\right)th \text{ value} = 59\\
Q_2 &= \left(\frac{2\times 96}{4}\right) = 48th \text{ position} = 98\\
Q_3 &= \left(\frac{3\times n}{4}\right)th = \left(\frac{3\times 96}{}\right)th \text{ position} = 72th \text{ position} = 108
\end{align*}

Note that the above data is already sorted. If the data is not sorted, we first need to arrange/sort it in ascending order.

Quartiles for Gruoped Data

One can also compute the quantiles for the following grouped data, hence the quartiles.

ClassesfxC.B.CF
65-84974.564.5-84.59
85-1041094.584.5-104.519
105-12417114.5104.4.5-124.536
125-14410134.5124.5-144.546
145-1645154.5144.5-164.551
165-1844174.5164.5-184.455
185-2045194.5184.5-204.560
Total60   

From the above-grouped data, we have 60 observations $(n=60)= \sum\limits_{i=1}^n = f_i = \Sigma f = 60$. The three quartile will be

\begin{align*}
\frac{n}{4} &= \left(\frac{60}{4}\right)th = 15th \text{ value}\\
Q_1 &= l + \frac{h}{f}\left(\frac{n}{4} – CF\right) = 84.5 + \frac{20}{10}(15-9) = 96.5\\
\frac{2n}{4} &= \left(\frac{2\times 60}{4} \right) = 30th \text{ value}\\
Q_2 &= l + \frac{h}{f}\left(\frac{2n}{4} – CF\right) = 104.5 + \frac{20}{17}(30-19) = 117.44\\
\frac{3n}{4} &= \left(\frac{3\times 60}{4} \right) = 45th \text{ value}\\
Q_3 &= l + \frac{h}{f}\left(\frac{3n}{4} – CF\right) = 124.5 + \frac{20}{17}(45-36) = 142.5\\
\end{align*}

Frequently Asked Questions about Quantiles

  1. Define Quartiles, Deciles, Percetiles.
  2. What are fractiles or Quantiles?
  3. How quantiles are computed for grouped and ungrouped data.

https://rfaqs.com

https://gmstat.com

MCQs Skewness Basic Statistics 9

The post is about MCQs Skewness Basic Statistics. There are 20 multiple-choice questions covering the topics of skewness, kurtosis, symmetrical distribution, and empirical relationship between mean, median, and mode. Let us start with the MCQs Skewness Basic Statistics Quiz.

Online MCQs Skewness and Kurtosis

1. In a symmetrical distribution, mean, median, and mode are:

 
 
 
 

2. The distribution is symmetrical if the moment coefficient of skewness $\sqrt{b_1}$ is

 
 
 
 

3. If in a distribution the left tail is longer than the right tail, then the distribution will be

 
 
 
 

4. A symmetrical distribution has a mean equal to 4. Its mode will be

 
 
 
 

5. The second moment about the mean is equal to

 
 
 
 

6. If the third moment about the mean is zero then the distribution is

 
 
 
 

7. For a moderately skewed distribution, which of the following hold

 
 
 
 

8. If the mean is less than the mode, the distribution is

 
 
 
 

9. In a symmetrical distribution, the mean is _________ mode

 
 
 
 

10. Bowley’s coefficient of Skewness lies between

 
 
 
 

11. The empirical relation between mean, median, and mode is

 
 
 
 

12. The values of mean, median, and mode can be

 
 
 
 

13. The shape of the symmetrical distribution is _______

 
 
 
 

14. A curve whose tail is longer to the right is called

 
 
 
 

15. If mean, median, and mode are all equal then distribution will be

 
 
 
 

16. If the mean is less than the mode, the distribution will be

 
 
 
 

17. In Uni-model distribution, if the mode is less than the mean, then the distribution will be

 
 
 
 

18. Which of the following is negatively skewed?

 
 
 
 

19. The distribution in which mean = 60 and mode = 50, will be ____________

 
 
 
 

20. For a positively skewed distribution

 
 
 
 

MCQs Skewness Basic Statistics with Answers

  • The shape of the symmetrical distribution is ————.
  • In a symmetrical distribution, mean, median, and mode are:
  • In a symmetrical distribution, the mean is ———— mode
  • A symmetrical distribution has a mean equal to 4. Its mode will be
  • If mean, median, and mode are all equal then distribution will be
  • The values of mean, median, and mode can be
  • The distribution in which mean = 60 and mode = 50, will be ————.
  • If in a distribution the left tail is longer than the right tail, then the distribution will be
  • If the mean is less than the mode, the distribution will be
  • In Uni-model distribution, if the mode is less than the mean, then the distribution will be
  • The empirical relation between mean, median, and mode is
  • Bowley’s coefficient of Skewness lies between
  • For a positively skewed distribution
  • For a moderately skewed distribution, which of the following hold
  • The distribution is symmetrical if the moment coefficient of skewness $\sqrt{b_1}$ is
  • A curve whose tail is longer to the right is called
  • If the mean is less than the mode, the distribution is
  • If the third moment about the mean is zero then the distribution is
  • Which of the following is negatively skewed?
  • The second moment about the mean is equal to
Statistics Help MCQs Skewness Basic Statistics

https://gmstat.com

https://rfaqs.com

Quiz Basic Statistics MCQs with Answers 8

The post is about Quiz Basic Statistics MCQs with Answers. There are 25 multiple-choice questions covering the topic measure of central tendency, measure of dispersions, data, variable, measurement of scale, statistical tables, sample, and population. Let us start with Quiz Basic Statistics MCQs.

Please go to Quiz Basic Statistics MCQs with Answers 8 to view the test

Quiz Basic Statistics MCQs with Answers

  • Which of the following describes the middle part of a group of numbers?
  • The sum of the deviations about the mean is always:
  • The middle value of an ordered array of numbers is the
  • Which of the following is not a measure of central tendency?
  • Which of the following divides a group of data into four subgroups?
  • If the standard deviation of a population is 9, the population variance is:
  • The weight of students in a college/ school is a
  • Population census is conducted through
  • The life of a TV picture tube is a
  • Which of these represents qualitative data
  • The branch of statistics concerned with the procedure and methodology for obtaining valid conclusions is called
  • Statistics are collected in a ———–.
  • Issuing a new ID card is an example of
  • Statistical Tables have at least ———–.
  • The students divided into different groups according to their intelligence and gender will generate
  • If the arithmetic mean is 82 and the median is 78 then the appropriate value of the mode will be
  • In a Statistical table, row captions are called ———–.
  • The data must be arranged before computing the
  • The geometric mean of a series of 4 items is 10.2. The product of all the items shall be:
  • Standard deviation is independent of change of ———–.
  • The positive square root of the variance of a distribution is known as ———–.
  • The most central value of an arrayed data is called ———–.
  • The mode of 2, 9, and 7 is
  • In classification, the data are arranged according to ———–.
  • The population mean $\mu$ is called ———–.
Quiz Basic Statistics MCQs With Answers

https://gmstat.com

https://rfaqs.com