Modified Box Plots

Introduction to box plots:

In statistics a box plot can be defined as a diagram explains the summary of data sets. For easy understanding the plots are modified in such a manner that a common man can examine the box plots. Generally creation of box plots involves in finding out the parameters such as median, quartiles, greatest and least observations.

Modified box plots explains the step by step procedure for creating box plots and with the help of it we can evaluate where the observations are concentrated and where they are spread out. As modified box plots are simple to understand they are widely used by many persons in different profession.

Examining Box Plots

Examining box plots is nothing but a process of reading the elements of the plot.

Actually, there are several parameters are available. Namely,

Median
Quartiles
Minimum and maximum values


The following figure shows a typical modified box plots.

Modified box plots



Median:

The median is the value that divides a data set into two parts and it is denoted as Q2.

Quartiles:

As said earlier the median divides the data set into two parts, the first part is known as Lower quartile and later is called as upper quartile.

These quartiles are considered as separate data sets and the median is evaluated.

Lower median is denoted as Q1 whereas upper median is denoted as Q3.

Maximum & Minimum values:

The greatest and the least values of the data set are called as maximum and minimum values respectively.

Outliers:

Upper outlier is the value which is more than 3/2 times of upper quartile and Lower quartile is the value which is less than 3/2 times of lower quartile.Is this topic Greatest Common Factor Word Problems hard for you? Watch out for my coming posts.

Example

Consider the following data set.

Data set => (12, 4, 9, 66, 8, 53, 37)

Solution:

Step 1:

Rearrange the data set in order.

(4, 8, 9, 12, 37, 53, 66)

As there are 7 elements in the data set. The fourth element lies exactly in the center.

So, median Q2 = 12

Step 2:

Quartiles:

Lower quartile = (4, 8, 9)

Median of lower quartile Q1= 8

Upper quartile = (37, 53, 66)

Median of the upper quartile Q3= 53

Step 3:

Minimum value= 4

Maximum value= 66

So, min=4, Q1=8, Q2=12, Q3=53, max=66

Step 4:

Draw a straight line which extends in both directions.

Step 5:

Draw the modified box plots using the parameters.

Modified box plot

Solving Basic Statistics Problems

Introduction to solving basic statistics problems:   

Everyday we come across a wide variety of information’s in the form of facts, numerical figures, tables, graphs, etc. These figures are the numerical data collected with a definite purpose, are called data. Data is a word in a plural form of the Latin word datum. Every part of our lives utilizes data in one form or the other. This extraction of significant information is studied in a branch of mathematics called Statistics.

Solving Basic Statistics Problems-statistics Definition

Basically the word ‘statistics’ appears to have been taken from the Latin word ‘status’ meaning ‘a state (related to a political)’. In its origin, statistics is simply the collection of data on different aspects of the life of people, useful to the State. Basic Statistics deals with collection, organization, analysis and Interpretation of data. The word ‘statistics’ has held different meanings in different contexts, so that the calculation process is based on the meaning and data’s.
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Solving Basic Statistics Problems

Example for basic stastistics problems:

Consider the marks obtained by 10 pupils in a mathematics test as given below:

55 36 95 73 60 42 25 78 75 62

The data in this form is called raw data.

By looking in this way, can you find the highest and the lowest marks?

Did it take you some time to search for the maximum and minimum marks? Wouldn’t it be a lesser amount of time consuming if these scores were arranged in ascending or descending order? So let us arrange the marks in ascending order as

25 36 42 55 60 62 73 75 78 95

Now, we can clearly watch that the lowest marks are 25 and the highest marks are 95.

The difference of the highest and the lowest marks in the basic stastistics data is called the range of the data. So, the range in this case is 95 – 25 = 70.

Arrangement of data in ascending or descending order can be quite time consuming, particularly when the number of observations in an experiment is large.

Variance of a Data Set

Introduction to variance of a data set:

Data is nothing but a set of information or collection of facts. We can analyze the data’s with the help of charts and graphs. The method of collecting or preparing the data’s is the data collection.
Variance is the term that  explains how average value of the data set vary from the measured data.

Variance of Data Set:

Following are the steps involved in finding the variance of data set:    

Step 1: First step is to find the Average, that is the arithmetic mean for the given set of numbers.

Mean = Sum of all values / Total number of elements

Step 2: To find the sum of squares for all the given values in the given data set.

Step 3: Divide the result we obtained in Step 2 over the total number of elements in the given data set.

Step 4: Find the difference between the square of mean and the result we obtained from step 3

Step 5: For finding the variance, take the square root of the number that  we obtained as a result from step 4.

Thus, using the above steps, we can easily calculate the variance of the given data set.

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Solved Examples for Finding Variance of a Data Set:

Ex1: Find the variance  of the given data set: { 5, 6, 9, 13, 19 }

Sol:

Step 1: Arithmetic mean             =  (5 +6 + 9+ 13 + 22) / 5

= 55 / 5

=  11

Step 2: Sum of squares               =  52 + 62 + 92 + 132 + 222

=  25 + 36 + 81 + 169 + 484

= 795

Step 3: Sum of squares / Total number of elements

= 795 / 5

= 159

Step 4: Result from Step 3 – square of mean value

= 159 – 112       

= 159 – 121

= 38

Step 5: Variance                        = square root ( 38 )

= 6.16

The variance of the given data set is 6.16

Ex 2: Find the variance  of the given data set: { 2, 3, 4, 5,16 }

Sol:

Step 1: Arithmetic mean             =  (2 +3 + 4+ 5 + 16) / 5

= 30 / 5

=  6

Step 2: Sum of squares               =  22 + 32 + 42 + 52 + 162

=  4 + 9 + 16 + 25 + 256

= 310

Step 3: Sum of squares / Total number of elements

= 310 / 5

= 62

Step 4: Result from Step 3 – square of mean value

= 62 – 62       

= 62 – 36

= 26

Step 5: Variance                        = square root ( 26 )

= 5.1

The variance of the given data set is 5.1

Percentile Definition Statistics

Introduction to percentile definition statistics:

The definition of percentile is dividing the data into 100 equal parts in statistics. We can also refer the percentile as quartiles. The percentile formula of statistics is using the samples. In statistics, the percentile is determined from ordered data. Now we are going to see about percentile definition statistics.

Explanation for Percentile Definition Statistics

Definition of percentile in statistics:

The percentile definition is the position of score that is part of scored values. In other words, it indicates the particular frequency percentage that is percent of scored values. It defines the score position in statistics.

Percentile formula in statistics:

The individual population is calculated by percentile and the statistics use the formula in different forms. They are,

If the ‘x’ values are used in statistics means we can use the percentile formula as `(B + 0.5E)/n` x 100.
If we does not use the ‘x’ values means we can use the percentile formula as` (Number of below x values)/n` x 100.
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More about Percentile Definition Statistics

Example problems for percentile definition statistics:

Problem 1: The scores of student are 22, 24, 25, 27, 30, 31, 32, 33, 33, 34, 36, 37, 39, 40, 42, 44, 45, 47, 50, 52.

Solution:

Total number of scores is 20 and number of below 33 is 7.

Percentile is `(B + 0.5E)/n` x 100 =` (7 + 0.5(2))/20` x 100 = `8/20` x 100 = 40.

Percentile of score 33 is 40.

Problem 2: Student scores are 12, 14, 15, 16, 17, 19, 21, 21, 22, 24, 26, 27, 29, 30, 31, 34, 35, 37, 38, 40. What is the percentile value of 21?

Solution:

Total number of score is 20 and the below 21 is 6.

The calculation for percentile is` (B + 0.5E)/n` x 100.

P = `(6 + 0.5(2))/20` x 100 = `7/20 ` x 100 = 35.

Percentile of score 21 is 35.

Exercise problems for percentile definition statistics:

1. The score of Alex is below 77 is 5 out of 20. What is the percentile value?

Solution: Percentile of Alex is 25.

2. Student’s scores are 2, 3, 4, 4, 6, 7, 9, 10, 12, 13. Determine the percentile of score 4.

Solution: The score 4 is at 30th percentile.

Decimal Place Value System

Introduction:
Decimal place value system is slightly differing from normal place value for numbers. This decimal place value system has a decimal point and numbers on both sides of decimal point. There are different names for the numbers before and after the decimal system. We are going to see how the place value system for decimals.

Explanation to Decimal Place Value System:

General representation of decimal place value system:

Let us see the place value system for the decimal 123456 . 789

Before the decimal point:

123456

6 - ones

5 - tens

4 - hundreds

3 - thousands

2 - ten thousands

1 - hundred thousands

After the decimal point:

789

7 -tenths `1/10`

8 -hundredths `1/100`

9 - thousandths `1/1000`

The above number 123456.789 can be obtained as,

(1 x 100000) + (2 x 10000) + (3 x 1000) +(4 x 100) + (5 x 10) x (6 x 1) + ( 7 x `1/10`) + (8 x `1/100` ) + (9 x `1/1000`)

Example Problems to Decimal Place Value System:
Example: 1

Which of the following is the place value of 5 in decimal 9.5?

a) Ones

b) Tenths

c) Hundreds

d) Decimal point

Solution:

Given, 9.5

5 lie after the decimal point. So, the place value of 5 is tenths.

Answer: b

Example: 2

Which of the following is the place value of 9 for decimal 98.42?

a) Tenths

b) Ones

c) Hundreds

d) Hundredths

Solution:

9 lie before the decimal point.

98

8 – ones

9 – hundreds

Answer: c

Example: 3


How will you write the expanded form the decimal 3.36?

Solution:

Given 3.36

Before the decimal point: 3

3 - Ones

After the decimal point: 36

3 - tenths

6 - hundredths

3.36 = (3 x 1) + (3 x `1/10`) + ( 6 x `1/100`)

Practice Problems to Decimal Place Value System:
Problem: 1

Which of the following is the place value of 8 in decimal 2.08?

a) Ones

b) Tenths

c) Hundredths

d) Hundreds

Answer: c

Problem: 2

Which of the following is the place value of 1 in decimal 0.001?

a) Ones

b) Thousandths

c) Hundredths

d) Hundreds

Answer: b

Algebra Ratio and Proportion

Introduction of algebra ratio and proportion:

Define of algebra ratio:

In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient

Define of algebra proportion:

If the two or more Ratio Proportion encompass all of the quantities in a particular situation.

Source: Wikipedia


Definition of Algebra Proportion:

The ratio `25/5` may be simplified and written as `5/1`.

`25/5`=`5/1`

An equation that states that two ratios are equal is called an algebra proportion. The preceding algebra proportion may also be written in the form 25:5=5:1.

Concept of algebra proportion:

A proportion is an equation that states that two ratios are equal: `m/n`=`p/q`  or  m:n=p:q (provided n?0 and q?0)

Each term of proportion is given a special name according to its position in the proportion.

`m/n`=`p/q`

Where

m is first proportional

n is second proportional

p is third proportional

q is fourth proportional

the pair of terms that form the 1st and 4th  proportional’s are called as the extremes of a proportion; the 2rd  and 3rd proportional of a proportion are called the means of a proportion.

The pair of terms m and p are called as the extremes of the proportion; the pair of terms n and q are referred to as the means of the proportion.

Example for algebra proportional:

Find the first proportional if the remaining terms of a proportion are 1,3 and 9.

Solution:

Let x=first proportional. Then

x/1=3/9

x=`1/3`

x`xx`3=`1/3``xx`3

3x=1

x=`1/3`

The first proportional is `1/3`.

Sometimes we may be able to work with lesser numbers by simplify an arithmetic ratio in the innovative proportion before cross-multiplying.
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Definition of Algebra Ratios:

The ration of two numbers a and b(b?0) is the quotient of the numbers. The numbers a and b referred to as the terms of the ratio.

Example problem for algebra ratios:

Example 1:

Solve `25/125`

Solution:

=`1/5`    (numerator and denominator are divided by 25)

=`1/5` 

Answer is `1/5`or 1:5

Example 2:

Solve`(30/15)``xx``(12/2)`

Solution:

=`(30/15)``xx``(12/2)`

= `360/30`(numerator and denominator are divided by 10)

=`36/3`  (numerator and denominator are divided by 3)

=12 or `12/1`

Answer is `12/1`  or 12:1

Absolute Values Algebra Help

Introduction to absolute values algebra help:

Absolute values in algebra help are nothing but we are going to get the help on the absolute values in algebra. In this we are going to solve the problems based on absolute value rules. For example the absolute values of the function or numbers are always positive. So if we solve the absolute value functions we will get the two values for the absolute value functions. We will see some example for absolute values in algebra.

Example Problems for Absolute Values Algebra Help:

Absolute values algebra help example 1:

Solve the variable x where |4x + 2| = |2x + 3|

Solution:

The given absolute value function is |4x + 2| = |2x + 3|

From this we can divide this into (4x + 2) = 2x + 3 ………. (1) – (4x + 2) = 2x + 3 ………… (2)

Equation 1:

(4x + 2) = 2x + 3

Add - 2 on both sides

4x + 2 – 2 = 2x + 3 - 2

4x = 2x + 1

Add – 2x on both sides

4x – 2x = 2x + 1 – 2x

2x = 1

So x = 0.5

Equation 2:

-(4x + 2) = 2x + 3

-4x -2 = 2x + 3

Add + 2 on both sides

-4x - 2 + 2 = 2x + 3 + 2

-4x = 2x + 5

Add – 2x on both sides

-4x – 2x = 2x + 5 – 2x

-6x = 5

So x = - `(5 / 6)`

We will see some more examples for absolute values in algebra. It is better to getting help on absolute values in algebra.

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Absolute Values Algebra Help Example 2:

Solve the variable x where |2x + 1| = |x - 1|

Solution:

The given absolute value function is |2x + 1| = |x - 1|

From this we can divide this into (2x + 1) = x - 1 ………. (1) – (2x + 1) = x – 1 ………… (2)

Equation 1:

(2x + 1) = x – 1

Add - 1 on both sides

2x + 1 – 1 = x – 1 - 1

2x = x - 2

Add – x on both sides

2x – x = x – 2 - x

So x = -2

Equation 2:

-(2x + 1) = x - 1

-2x -1 = x - 1

Add + 1 on both sides

-2x -1 + 1 = x -1 + 1

-2x = x + 0

Add – x on both sides

-2x –x = x - x

-3x = 0

So x = 0

These are some of the examples for absolute value problems in algebra. This is better to getting help on absolute values.