Histogram

Frequency distributions can be represented graphically using different types of graphs. Histogram, frequency polygon, frequency curves, and Ogive are the different types of graph which represents frequency distribution.
Histogram is an area diagram. A set of rectangles with bases along the intervals between class boundaries and with areas proportional to the frequencies in the corresponding classes define a histogram. In such a representation the rectangles are all adjacent, since the bases cover the intervals between class boundaries, not class limits. With equal class intervals the heights of rectangles will be proportional to corresponding frequencies, while for unequal classes they will be proportional to corresponding frequency densities.
A histogram is a graphical display of data using bars of different heights. Total area of the rectangles in a histogram represents the total frequency. A histogram with more number of class intervals is more effective in depicting the structure of the frequency distribution.

Definition of Histogram states that a two dimensional frequency density diagram is called as a histogram. The histograms are diagrams which represent the class interval and the frequency in the form of a rectangle. There will be as many adjoining rectangles as there are class intervals.

The following steps are to be followed to construct a histogram.

(1) Mark class intervals on X-axis and frequencies on Y-axis.

(2) The scales for both the axes need not be the same.

(3) Class intervals must be exclusive. If the intervals are in inclusive form, convert them to the exclusive form.

(4) Draw rectangles with class intervals as bases and the corresponding frequencies as heights.

(5) The class limits are marked on the horizontal axis and the frequency is marked on the vertical axis. Thus, a rectangle is constructed on each class interval.

(6) If the intervals are equal, then the height of each rectangle is proportional to the corresponding class frequency.

(7) If the intervals are unequal, then the area of each rectangle is proportional to the corresponding class frequency.

Given below are some of the examples on histogram.

Example 1:

Draw a histogram for the following data:

Class Interval	Frequency
0 - 5	4
5 - 10	10
10 - 15	18
15 - 20	8
20 - 25	6

Suggested Answer:

Note:

In the above example, the intervals are exclusive. Now, let us consider an example with inclusive intervals.

Example 2:

The daily wages of 50 workers, in rupees, are given below:

In table (a), the class intervals are inclusive. So we convert them to the exclusive form as shown in table (b).

Wages (Rs)	Frequency
51 - 60	4
61 - 70	12
71 - 80	8
81 - 90	16
91 - 100	4
101 - 110	6

Wages (Rs)	Frequency
50.5 - 60.5	4
60.5 - 70.5	12
70.5 - 80.5	8
80.5 - 90.5	16
90.5 - 100.5	4
100.5 - 110.5	6

Suggested Answer:

Note:

(i) The class intervals are made continuous and then the histogram is constructed.

(ii) A kink or a zig - zag curve is shown near the origin. It indicates that the scale along the horizontal axis does not start at the origin.

(iii) The horizontal scale and vertical scale need not be the same.

Histogram	Bar Graph
1. It consists of rectangles touching each other	It consists of rectangles normally separated from each other with equal space
2. The frequency is represented by the area of each rectangle	The frequency is represented by height. The width has no significance.
3. It is two dimensional	It is one dimensional
	It is used as a virtual aid to represent data

Uniform random distribution is the one of the important topic in the probability distribution chapter in mathematics subject. The simplest distribution is the uniform random distribution; The uniform random distribution is suitable for most sensitivity testing and is selected by default. And it is classified into two types; one is discreet uniform distribution and another one is continual uniform distribution.

Uniform Random Distribution

In order to analyses numerical data, it is necessary to arrange them systematically. An arrangement of data in a systematic order is called a uniform distribution. A uniform distribution, sometimes called as a rectangular distribution, in this distribution that has the constant Probabilities occurred.

Types of Uniform random distributions:

Uniform random distributions are classified as two types, they are

1. Continual uniform distributions, and

2. Discrete uniform distributions.

Continuous & Discrete Uniform Distributions:

Continual uniform distributions:
It is a statistical distribution for which the variables take on continual range. There are certain phenomenon which by the lack of precision in measurement are not capable of exact measurement.

Example: weight, height, temperature, age, etc.,
Such a series are called as continual distributions.

Discrete uniform distributions:
It is also the statistical distribution where the variables can take on only discrete values. A discrete distribution is formed from items which are capable of exact measurement.
A discrete distribution with probability function p(x_k) defined over, k = 1,2...N., Has distribution function.

D (x_n) = `sum_(k=1)^n` p(x_k)

and population mean, is `mu` = 1 / N `sum_(k=1)^n`x_{k P}(x_k)

Ex: we can count the number of Parsons salaries are exactly Rs 100 p.m, Rs 105 p.m., or Rs 110 p.m. Other examples of discrete variables are the number of children in a family, goals scored in foot ball matches.

Uniform Random Distribution General Formula:

The general formula of probability density function of the uniform ramdom distribution function is defined as follows:

f(x) = 1 / B-A for A`<=` x`<=` B

Where A is the location parameter and (B - A) is the scale parameter. The case where A = 0 and B = 1 is called the standard uniform random distribution.

The equation of the standard uniform random distribution is

f(x) = 1 for 0 `<=` x `<=` 1.

These all are important in the Uniform random distributions.