Frequency Distribution Tables

Frequency distribution:

The frequency distribution represents the number of observations with in the interval These are  graphically or tabular format.But the intervals must be mutually exclusive and exhaustive,and these are used with in a statistical context.The frequency distribution is a table it contains classes and its representing frequency.The class means quantitative or qualitative type in which the placed of raw data.The data is two types .1)Primary data 2)Secondary data.Primary data is collected by who are the investigator,some times an investigator uses another investigator of primary data.This type of data is called secondary data.The secondary data grouping and presenting in form of table.These type of tables are called Frequency  distribution tables.The data form is in the table shows below.

Data Range:

It is Highest value-lowest value

Class width

Range/ desired number of classes

Upper/lower class limit – upper/lower class Limit of next class

Upper class boundary – lower class Boundary

Class midpoint

Xm   = ( Lower bound + Upper bound)/2

Or

Xm = (Lower Limit + Upper Limit) /2


Frequency Distribution table:


.We can represents The frequency distribution table in number of ways.

1)Group frequency table or graph(polygon,chart)

2)Regular frequency table or Bar graph(Histogram)

When the data is from nominal or ordinary scale then we use Bar graph.

When the data measurements consists more categories than the listed in regular table then we use Group frequency distribution.


Important Notes:


1)The grouped frequency distribution table must use when the range of scores is large, causing a regular frequency table to have too many entries in the score categories (X column). The guidelines for a group frequency table include   approximately 10 rows in the table

2)  The Interval width of 2,5,10,20,50,100 should be used  it depends on the number of rows chosen

3)  Each interval of the  first (lowest) value of should be a multiple of the interval width

4) Note all intervals without missing any, The top interval must contain the highest observed X value and the bottom interval should contain the lowest observed X value.

Example:

For  the given data below into a frequency distribution table and show percentage of each category.
8, 9, 8, 7, 10, 9, 6, 4, 9, 8,
7, 8, 10, 9, 8, 6, 9, 7, 8, 8



x    f    ρ
10    2    0.1
9    5    0.25
8    7    0.35
7    3    0.15
6    2    0.10
5    0    0
4    1    0.05

Solving Irrational

solving irrational numbers:

Irrational numbers in difference to rational numbers aren’t presented a fraction of the shape: m / n,  where m and n are integers. There are numbers of a new type, which are calculated with any accuracy, but can’t be changed by a rational number. They can appear as outcome of geometrical measurements,

for example:

- a ratio of a square diagonal duration to its side length is equal to √2.

- a ratio of a circumference duration to its diameter length is an irrational number  Pi



Where can you find special irrational numbers?(solving irrational)


The answer to this depends on what you consider 'special'.  Mathematicians have proved that definite special numbers are irrational, for example Pi and e.  The number e is the base of natural logarithms.   It is irrational, just like Pi, and has the value 2.718281828459045235306....

It's not easy to just "come up" with such special numbers.  But you can easily discover more irrational numbers after you've found that most square roots are irrational.  For example, what do you think of √2 + 1?  Is the result of that addition a rational or an irrational number?  How can you know?  What about other sums where you add one irrational number and one rational number, for example √5 + 1/4?

You can also add two irrational numbers, and the amount will be many times irrational.  Not always though; for example, e + (-e) = 0, and 0 is rational even though both e and -e are irrational.  Or, take 1 + √3 and 1 - √3 and add these two irrational numbers - what do you obtain?

Or, increase/separate an irrational number by a rational number, and you get an irrational number.  For example, √7/10000 is an irrational number that is moderately close to zero.  Yet another possibility to discover irrational numbers is to multiply square roots or other irrational numbers.  Sometimes that outcome in a rational number though (when?).  Mathematicians have also calculated what happens if you raise an irrational number to a rational or irrational power.

Yet more irrational numbers begin when you take logarithms, or calculate sines, cosines, and tangents.  They don't have any particular names, but are just called "sine of 70 degrees" or "base 10 logarithm of 5" etc.  Your calculator will give you decimal approximation to these.

Examples of Irrational Number


`Pi`,  √10 Are few examples of irrational numbers.

Solved Example on Irrational Number

Identify the irrational number.

Choices:
A. 0.3
B. 3 / 5
C. √6
D. √25
Correct Answer: C

Solution:
Step 1: The value of  is 2.4494897427831780981972840747059...
Step 2: The value of  is 5.
Step 3: The value of  is 0.6.
Step 4: Among the choices given, only  is an irrational number

FRACTIONS UNLIKE DENOMINATORS

A fraction  is a number that can represent part of a whole. An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole.

How to Add Fractions with Unlike Denominators:

To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction.

Consider adding the following two quantities:

3/4 + 2/3

First, convert  3/4  into twelfths by multiplying both the numerator and denominator by three3/4 x 3/3 =9/12

Note that  3/3 is equivalent to 1. which shows that  ¾ is equivalent to the resulting  9/12.

 Secondly, 2/3convert  into twelfths by multiplying both the numerator and denominator by four:2/3 x 4/4 = 8/12. Note that 4/4  is equivalent to 1, which shows that  2/3  is equivalent to the resulting 8/12 .

9/12 + 8/12 = 17/12.


Simplify the Fraction

Example: Find the Sum of 2/9 and 3/12

Determine the Greatest Common Factor of 9 and 12 which is 3

Either multiply the denominators and divide by the GCF (9x12=108, 108/3=36)

OR - Divide one of the denominators by the GCF and multiply the answer by the other denominator (9/3=3, 3x12=36)

Rename the fractions to use the Least Common  Denominator(2/9=8/36, 3/12=9/36)

The result is 8/36 + 9/36

Add the numerators and put the sum over the LCD = 17/36

Simplify the fraction if possible. In this case it is not possible.

Example:

Find. 3/4  - 2/3

Solution:

Since the fractions have different denominators, they cannot be subtracted until they have the same denominators.

We can express both fractions as twelfths.

3/4 = 4/12 and 2/3 = 8/12

9/12 - 8/12 = 1/12.

Example:

Find

 5/6 + 2/3

Solution:

Since the fractions have different denominators, they cannot be added until they have the same denominators.

We can express both fractions as sixths.

2/3 = 4/6

5/6 + 3/6 = 8/6

Graphing Inequalities

In this article we are going to discuss about graphing inequalities concept.A coordinate graph is called as the Cartesian coordinate plane. The graph contains a couple of the vertical lines are called coordinate axes. The vertical axis of the y axis value and the horizontal axis value is the x axis value. The points of the intersection of those two axes values are called the origin of coordinate graphing pictures. In this graph linear equation is in the form of y = mx + c. I

Graph of inequalities

Below you can see some examples on graphing inequalities -

Problem 1 :

Solving the system of inequalities for two variables equation use graphing method 7x – 6y < 5 and draw the graph for a given inequalities.

Solution:

We find the plotting points of the given inequality equation. The given equation is

            7x - 6y < 5

We are going to find out the plotting points for a given equation. In the first step we are going to change equation in the form of y = mx + c, we get the following term

                7x - 6y < 5                  

                  7x < 6y + 5

In the next step we are multiply the value -1 to equation (1), we get

            -7x > - 6y - 5

               6y > 7x – 5

In the above equation we put x = -1 we get

            6y > -7 - 5

            y > -2

In the above equation we put x = 2 we get

            6y > 14 – 5

             y > 1.5

We draw a graph for given equation. We get





Solving the system of inequalities for two variables equation use graphing method 4x – 3y > 2 and draw the graph for a given inequalities.

Solution:

We find the plotting points of the given inequality equation. The given equation is

            4x - 3y > 2

We are going to find out the plotting points for a given equation. In the first step we are going to change the above equation in the form of y = mx + c, we get

      4x - 3y > 2

            4x > 3y + 2                 

In the next step we are multiply the value -1 to a above equation, we get

               -4x < -3y - 2

                   3y < 4x - 2

In the above equation we put x = -1 we get

            3y < -4 - 2

             y = -2

In the above equation we put x = 2 we get

            3y < 8 - 2

             y < 2

We draw a graph for given equation. We get

The graphical concept of x intercept and y intercepts is Very simple. The x intercepts are the points where the graph crosses the x-axis, and the y-intercepts are the points where the graph crosses the y-axis. The problems start when we deal with intercepts algebraically.

The X and Y intercept are drawn in the graph to find the slope of the graph from the given equation.

Then, algebraically,

an x intercept is a point on the graph where the point y =  zero, and
A y intercept is a point on the graph where the point x = zero.

Procedure for finding X and Y intercept

x Intercept

Step1: For finding the X intercept we need to arrange the given equation in the slope intercept form.

Step2: The slope intercept form is    y = mx + c

Step3: After this we substitute y=0.

Step4: The equation becomes mx + c = 0, now we need to add/subtract constant C on both sides.

Step5:  Then divide the constant by the slope.  x = c/m.

y Intercept

Step1: For finding the y intercept we need to arrange the equation in slope intercept form.

Step2: After this, substitute x = 0

The equation becomes y= c, this is the Y intercept.


y and x intercept for 2x+y=8.


2x+y=8

The given equation is in the form of slope intercept form i.e. Y = mX + C

To find X Intercept, put y = 0

0 = -2x + 8

Subtract 8 on both sides,

0 – 8 = -2x + 8 – 8

-2x = -8

Divided 2 on both sides

-2x/2 = `-8/2`

-x = -4

Here the x intercept is 4.

To find the Y Intercept, put x = 0

y = 0 + 8

y = 8

Here the y Intercept is 8.

Using this we can plot the graph.

Vector Components

Here, we are going to discuss about the components of vector, generally vector has both the direction and the magnitude. Components, we mean as the parts of vectors, otherwise we can tell the two perpendiculars that added together means the components of the vector will give the original vector .The concept of the component vector is also the same as the rule of the vector addition .Vectors are denoted by the arrow look like as the below`->`

Rules of Vector Components

Here, we are following some rules in the components of the vectors those are given below:

• Components should always be perpendicular sometimes we called, the orthogonal components.
• The component s of the vector may be in any axis (x and the y axis) we called the horizontal or the vertical dimension.
• The direction of the components is look like the head to tail, so that we can add that vector. (because it is the components)
• So the components of the vectors are mutually independent .It will be shown through the following figure.
• If we are adding those x and y vectors we can get the resultant vector.

Examples on components of vectors

Example 1: Calculate the components of vectors along with the coordination of the position vector of P(− 4, 3)

Solution: Here, the position vector is given below `vec(OP)` = −7`veci` + 13`vecj` ,So the Component of the position vector `vec(OP)`  along x-axis is given as  − 7`veci ` . the component of  the position vector `vec(OP)`along x-axis is the  vector with the  magnitude of the 7 and here, we have to mention the direction , its direction is along the negative direction of x-axis ,because -7 is given.

So, the Component of the position vector  `vecOP` along  the y-axis is 13. The component of the position vector which as `vec(OP)` along y –axis is a vector with the  magnitude  of 13,here the direction of the vector have the positive. So,it will be the positive direction along the y-axis.

Example 2: Calculate the x-component of  d2  using the sin function the opposite value is given as 35 degree and the hypotenuse value is 33m?

Solution : Sin `theta` = opposite/hypotenuse

Sin 35 = d2x/23m

d2x = (33m)(sin 35)

d2x = (33m)(0.5736)

d2x = 18.9288m

We can check that the value, whether it is positive or negative directions. In this problem This x-component points to the right so, it is positive.

d2x = 18.928756m is the answer.

Bar Graph Examples

Students can learn about Bar Graphs from the Bar Graph Examples. They can get help with plotting bar graphs from the online Statistics tutors.

A bar chart or bar graph is a chart with rectangular bars with lengths proportional to the values that they represent. The bars can also be plotted horizontally.

Bar charts are used for plotting discrete (or 'discontinuous') data i.e. data which has discrete values and is not continuous. Some examples of discontinuous data include 'shoe size' or 'eye color', for which you would use a bar chart. In contrast, some examples of continuous data would be 'height' or 'weight'. A bar chart is very useful if you are trying to record certain information whether it is continuous or not continuous data.


Examples of bar graphs


Definition for bar graph:

Bar graph can be either vertical or horizontal. There may be just one bar or more than one bar for each interval. Sometimes each bar is divided into two or more parts. The data is made up of more than one category. In this section, you will work with a variety of bar graphs. Be sure to read all titles, keys, and labels to completely understand all the data that is presented. The Bar graph examples cover all the types of bar graphs.

Examples of bar graphs:

Bar graph Examples 1:

The results for last year high school math grades have been calculated. How many more 9th graders made A’s in math than 11th graders?

Solution:



Step1: Looking under the category of A’s, locate the graph for 8th graders. The interval on the left side of the bar graph reads 75 at the highest points on the graph for 8th graders. That means 75 8th graders made an A in math class.

Step 2:  Now do the same thing in step 1 again except the time instead of looking at the graph for 8th graders. We look at the graph for 10th graders. The interval on the left side of the bar graph reads 50 at the highest point on the graph for 10th graders. That means 50 10th graders made an A in math class.

Step 3:  Subtract the 50 10th graders from the 75 8th graders. There is an excess of 25 students. This means that 25 more 8th graders made on A in math than 10th graders.


Example for bar graphs 2:


Bar graph Examples 2 :

Graph show the number of rainy days in the months since May to September. How various rainy days are here in July, August, and September altogether?

Solution:



Find the number of rainy days in every of the three months and add them alone

Step 1:            Find the months scheduled along the horizontal axis.

The months are scheduled in a row from May to September.

Step 2:           Locate the bars for July, August and September.

Step 3:           Find the number of rainy days for July, August and September. Move up the bar from the label “July” until you approach to the top of the bar.  After that move to the left to discover the number of rainy days. The answer is 7, since the top of the bar is halfway among the 6 and 8. Do the same for August and September.

July – 7 rainy days

August – 3 rainy days

September – 6 rainy days

Step 4:           Add the three months:

7 + 3 + 6 = 16 rainy days

There are 16 rainy days in July, August and September.

Students can also avail help with Statistics homework problems involving Bar graph examples from the online tutors.