How to Find Quartiles?

What are Quartiles? 
Quartiles are used in descriptive statistics.  It is used to describe one-fourth of the given data. Quartiles refer to the 3 points, which divide the whole data taken for statistics into four equal parts.  Here is an overview about the three points or quartiles:

• The first quartile i.e. the first point will separate the first quarter of the data. It is also called as lower Quartile or 25th percentile.
• The second quartile is the median or the middle point, which divides the whole data into two halves. This can be termed as 50th percentile.
• The third quartile is the point that splits the highest quarter of the data from the data set. This is called as the upper quartile or the 75th percentile.

The data values till Q1 is the first quarter, the data values in between the points Q1   and Q2   is the second quarter, the data values in between the points Q2  and Q3  is the third quarter and the data values after Q3  is the fourth quarter. This is the way the three quartiles divide the whole data set into four quarters.

How to find Quartiles?
Now that you know what a quartile is, the next step is to know how to find quartiles.  For finding quartiles, we have to know where the three points or the quartiles lie on the given data set. In order to find quartile, sort the values in the given data set in the ascending order.

The quartiles or the points can be found out by using the formula: L is equal to y multiplied by (x divided by 100). Here, x refers to the percentile for the first, second and third quartiles respectively and y refers to the number of data values present in the given data set in which the data values are arranged in ascending order. L refers to the location of the quartile.

Examples of Quartiles
Let us consider a class of 20 students whose English marks have to be compared to find the number of students who are below average, average and excellent in English subject.
The marks obtained by the students are 80, 73, 64, 99, 67, 96, 87, 53, 43, 97, 78, 90, 69, 81, 40, 35, 100, 96, 93, and 98.

The above data is arranged in ascending order as: 35, 40, 43, 53, 67, 64, 69, 73, 78, 80, 81, 87, 90, 93, 96, 96, 97, 98, 99, and 100.

As per formula the 25th percentile will be 20 multiplied by 25 divided by 100 which will be 5.  As the value of L25   is 5, the number in the 5th location is the first quartile which is 67.  By similar calculations we get the 50th percentile or L50 i.e. 80 as the second quartile which is the 10th position and the third quartile is 96 which is in the 15th position or L75 in the data set. Based on the three quartiles, the four quarters are identified as highlighted below in red:

35, 40, 43, 53, 67, 64, 69, 73, 78, 80, 81, 87, 90, 93, 96, 96, 97, 98, 99, 100

Students within 67 marks are below average. Students who scored from 67 to 96 are average scorers and students scoring above 96 to 100 are excellent performers.

Know more about the statistics tutor. This article gives basic information about finding quartiles . Next article will try to cover more statistics help topics and its problems and many more. Please share your comments.

Absolute value of an integer

The absolute value of an integer is defined as the numerical value with out its sign. The absolute value of an integer is the distance between the number and zero on a number lineand is not considered which direction from zero the number lies.

Example:
The absolute value of |5| = 5
        |-5| = 5
How to Solve Absolute Value Equations
Solve the absolute value equation can be using the following steps.
1. First absolute value is isolated on any one side of the equation.
2. Then write two equations with out absolute symbol or bars.
3. First equation will fix the expression within the absolute value symbol equal to the given expression on the other side of the equal sign.
4. Second equation will fix the provided expression within the absolute value symbol equal to given expression on the other side of the opposite sign.
5. Finally solve the equation and get the solution.
Examples 1:
|2x-1| + 3 =6
First isolate the above equation
|2x-1| = 6-3
|2x-1| = 3
Then write in to two equations with out absolute symbol.
2x-1 = 3 2x-1 = -3
2x = 3+1 2x = -3+1
X = 2 x = -1
The solutions are {-1, 2}

Give some example to solve the Absolute value problems
Example 1:
|3(x+4)| = 24
3(x+4) =24 3(x+4) =-24
3x = 24-12 3x = -24-12
3x = 12 3x = -36
X = 4 x = -12
The solutions are{-12, 4}
Example 2:
6 / |x+3| = 3
First isolate the equation
3|x+3| = 6
3x+9 = 6 3x+9 = -6
3x = -3 3x = -15
X = -1 x = -5
The solutions are {-5, -1}
How to solve the inequalities with absolute value
To solve the inequalities with absolute value first isolate the equation on one side. Then the inequalities can be spilt in to two equations as positive and negative as per properties of the absolute value. Finally solve the equations to get the solution.
Example 1:
|x-1| = 2
X -1 = 2 x-1 = -2
X = 3 x = -1
The solution is -1 = x = 3
Example 2:
|3x+1| = 2x+3
3x+1 = 2x+3 3x+1 = -(2x+3)
3x-2x  = 3-1 3x + 2x = -3-1
X = 2 x = -4/5
The solution is -4/5 = x = 2
The absolute value inequalities

Here some of the absolute value inequalities can be explained with the absolute value equation and picture of real number line.

For example to graph the solution to |x|<5, the solutions of the absolute value of all the point should be less then 5 unit away from the zero. The solution of the absolute value inequalities are -5 < x < 5.

If |x| > 2, the solutions of the absolute value of all the point should be greater then 2 unit away from the zero.  The solution of the absolute value inequalities are -2 > x >

Introduction to Skewness

Skewness   can be defined as the asymmetry in the Distribution of data.The value on one side of the distribution tend to be further more than the “ Middle”   than the values on the other side.The Distribution is said to be skewed if the Mean and the Median falls at different points in a distribution. If the balance or centre of gravity is shifted to left then it is called positively skewed and if the balance or centre of gravity shifted to right   is called as negatively skewed.  Positively Skewed is more common than the negatively skewed.

Skewness Example:  In normal condition the distribution will be in Bell shape.  Whereas in skewness the distribution yields more either at left or right.Consider the following example:
The monthly wise AC machines sold in a year from January to December are as follows:
30,45,60,70,80,60,40,40,35,35,30,25.  The sale affected by climatic conditions. Peak sale is in May i.e.80.  Up to May the sale is progressive and from June it is gradually decreased.  If we plot a graph the picture will be skewed and represents Positively Skewed Distribution.

Interpreting Skewness: If the distribution is asymmetrical then it is called skewness.   If Mean is greater than the mode then the skewness will be positive.  If the difference is greater the skewness will be more.  The frequencies in the distribution as in the above example say the marks of students in a college is very high marks achieved by maximum students then the graph will be skewed towards left.  Then it is called negatively skewed.
• If Skewness is less than -1 or greater than + 1 the distribution can be interpreted as highly skewed.
• If Skewness is in between -1 to -1/2 or 1 to ½ the distribution is moderately skewed.
• If skewness is in between -1/2 to ½ then the distribution is approximately symmetric.

Measure of Skewness: Measure of skewness tells us the asymmetry of direction of frequency distribution. Skewness can be measured in absolute terms by taking the difference between Mean & Mode.  There are four important measures of Relative Measure of skewness namely:
• The Karl Pearson’s Coefficient  of Skewness
• The Bowley’s  Coefficient of Skewness
• The Kelly’s Coefficient of Skewness
• Measure of Skewness based on Moments
Skewness of Data: In a college, the marks scored by the student are as follows:
54,57,63,69,78,83,83,84,84,85,85,85,86,86,86,86,87,87,88,90
The average of marks = Mean = 84.55
Mode = the data value of highest frequency = 86 (4 times repeated)
Absolute Skewness is defined as the difference of Mode and Mean. Mathematically, it can be represented as
Absolute skewness = Mode - Mean
Histogram Skewness: As in the skewness of data, the data can be classified in to class intervals.
Class Interval          Frequency
50 - 60 2
60 – 70 2
70 – 80 1
80 – 90 15

Problem related to Arithmatic Progression

Which term of the AP : 21, 18, 15, . . . is – 81? Also, is any term 0? Give
reason for your answer.

Here is the step by step explanation to the problem,

Solving Arithmatic Progressions

Which term of the AP : 21, 18, 15, . . . is – 81? Also, is any term 0? Give
reason for your answer.

Here is the step by step solution to the problem,