Statistics:
The word statistics can be used as a plural sense and in a singular sense. In plural sense we use this word like statistics of prices, statistics of road accidents, statistics of educational institutions, etc. In all these examples the word statistics denotes a set of numerical data in the respective fields and most of the people usually use the word data instead of statistics. In singular sense the word statistics is defined as a discipline of science which deals with the collection, presentation, analysis and interpretation of numerical information.
Definition:
“It is the branch of science which deals with the collection, presentation, analysis and interpretation of numerical information”.
Population:
“Complete information of well defined group is called population” and the size of population is denoted by “N”.
There are two types of population finite population and infinite population. A population containing a finite and countable number of individuals is called a finite population, e.g. All the students of school, heights of all students of a college, all the books in a library etc. A population containing an infinite number of elements is called an infinite population, e.g. all fish in a lake, all trees of a country, all points on a line etc.
Sample:
“A sample is a part or a subset of a population or a representative part of a population is called a sample”. The size of sample is denoted by “n” e.g. the selected statistics book from library, the small group of some selected students from a college, the selected group of a few patients from a hospital.
Branch of statistics:
Statistics as a subject can divide into two parts descriptive statistics and inferential statistics.
Descriptive statistics:
Inferential statistics:
If the population is large or instead of population sample is given then we use inferential statistics to predict population information by the help of sample.
Importance of Statistics:
The world is becomingmore and more quantitative many professions depend on numerical measurements to make decision in the face of uncertainty, statistician use quantitative abilities, statistical knowledge and communication skills to work on many challenging problems.
Statistics assists in summarizing the larger sets of data in a form that is easily understandable.
Consciously or unconsciously statistics has entered in all branches of knowledge where a study of quantitative phenomenon is required. There is hardly any field whether it be commerce, industry, economics, biology, psychology, basic sciences, education, medicine, engineering, where statistician tools are not applicable.
Observation:
In statistics, an observation often means any sort of numerically recording of information whether it is a physical measurement such as height or weight, a classification such as heads or tails or an answer to a question such as yes or no.
Variables:
Any characteristic or quantity which varies individual to individual or object to object is called variables. A variable is usually denoted by the capital letter X or Y.
Types of variables:
Variables may be classified into quantitative and qualitative according to the form of the characteristic of interest.
Qualitative variable:
Quantitative variable:
Variables that have are measured on a numeric or quantitative scale. Ordinal, interval and ratio scales are quantitative. A country’s population, a person’s shoe size, or a car’s speeds are all quantitative variables. Variables that are not quantitative are known as qualitative variables.
OR
A quantitative variable is a variable that can be measured by a number, usually on a ratio scale, but at least on an interval or ordinal scale, such that less and more can be measured and determined. E.g. cars in a car park
Levels of Measurement
Measurement scales differ in their level of measurement. There are four common levels of measurement:
Nominal Scale
No ordering is implied, and addition/subtraction and multiplication/division would be inappropriate for a variable on a nominal scale. {Female, Male} and {Buddhist, Christian, Hindu, Muslim} have no natural ordering (except alphabetic). Occasionally, numeric values are nominal: for instance, if a variable was coded as Female=1, Male=2, the set {1,2} is still nominal.
Ordinal Scale:
An ordinal scale is a set of ordered values. However, there is no set distance between scale values. For instance, for the scale: (Very Poor, Poor, Average, Good, Very Good) is an ordinal scale. You can assign numerical values to an ordinal scale: rating performance such as 1 for "Very Poor," 2 for "Poor," etc, but there is no assurance that the difference between a score of 1 and 2 means the same thing as the difference between a score of and 3.
Interval Scales:
Interval scales are numerical scales in which intervals have the same interpretation throughout. As an example, consider the Fahrenheit scale of temperature. The difference between 30 degrees and 40 degrees represents the same temperature difference as the difference between 80 degrees and 90 degrees. This is because each 10 degree interval has the same physical meaning (in terms of the kinetic energy. Unlike ratio scales, interval scales do not have a true zero point.
Ratio Scale:
Types of Quantitative variables:
A quantitative variable may be classified as discrete or continuous.
Discrete variable:
A discrete variable is one that can take only a discrete set of integer or whole numbers. A discrete variable represents count data such as the number of persons in a family, the number of students in a class, etc.
Continuous variable:
A variable is called a continuous variable if it can take on any value fraction or integral within a given interval. A continuous variable represents measurement data such as the age of a person, the height, weight, temperature, etc.
Data and its types:
From a practical point of view the first step with which statistics deals is the collection of numerical data. These data are needed in different fields of human activity.
According to the sources, statistical data may be classified into two types, namely
Primary data:
Data which are collected for the first time for a specific purpose and are original in nature are known as primary data. It is unpublished data.
Collection of primary data:
First we make Questionnaire then we use some of the following methods to collect primary data:
Personal Interview
Telephonic survey
Through Post
Secondary data:
Collection of secondary data:
The secondary data may be obtained from the following sources:
Official
E.g. The publications of the statistical division, Ministry of finance, F.B.S, S.B.S, etc.
Semi Official
E.g. Newspaper, Internet, Books, Research Organization etc.
Presentation of data:
To put the collected data in such a way that one get more information in less time is known as presentation of data. There are two methods which may be used for the presentation of collected data.
Tabular Form (Frequency Distribution)
Graphical or Diagrammatic Presentation
Frequency Distribution:
A frequency distribution is a statistical table which shows the arrangement of data according to magnitude or size, either individually or in groups with their corresponding number of values side by side. The data presented in this form are also called grouped data.
Types of frequency distribution:
There are three types of frequency distribution
Categorical frequency distribution
District South Malir West East Central
No. of people(in lac) 15 35 20 25 30
Simple (ungroup) frequency distribution
Age 18 19 20 21 22
No. of students 5 10 15 10 5
Grouped Frequency distribution
Marks 1---5 6---10 11---15 16---20 21---25
Basic steps for constructing a grouped frequency distribution:
The following are some basic rules that should be kept in mind when constructing a grouped frequency distribution.
Determine the number of classes
k number of classes
There is no hard and fast rule for finding the exact number of classes. As a general rule the number of classes should be in between 4 and 15. The number of classes actually depends on the size of data. Therefore, we will select mostly the number of classes with our own judgment.
OR
There are some rules to determine number of classes
k = 1 + 3.3 log10 (N) or k = N total number of observations.
Determine the range
Range = r = maximum - minimum
Determine the class width
If ‘h’ is the width of class intervals then
Deciding the starting point
(i.e. lower limit of the first class) The starting point should be the min. value or less than the min. value of the data.
Deciding the nature of classes
When we have integer or whole number type of data then the classes should be in integer but if the data is on one decimal in nature then classes should be in one decimal in nature& so on.
Determining the remaining classes
Graphical presentation:
Following are the some graphical methods to represent data in a graphical form.
Scatter Plot
Simple bar diagram
Multiple & component bar diagram
Pie chart
Histogram
Frequency curve/ polygon
Ogive (cumulative frequency curve)
Exercise:
1. The following data represent the viscosity (friction, as in automobile oil) taken from 60 manufacturing batches (ordered from lowest viscosity to highest viscosity). Construct a frequency distribution of 5 classes and a percentage distribution?
12.6 12.8 13.0 13.1 13.3 13.3 13.4 13.5 13.6 13.7 13.7 13.7 13.8 13.8 13.9 13.9 14.0 14.0 14.0 14.1 14.1 14.1 14.2 14.2 14.2 14.3 14.3 14.3 14.3 14.3 14.3 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.5 14.5 14.5 14.5 14.5 14.5 14.6 14.6 14.6 14.7 14.7 14.8 14.8 14.8 14.8 14.9 14.9 14.9 14.9 14.9 14.9 2. Collected data about the cost of a meal per person from a sample of 50 city restaurants and 50 suburban restaurants are as follows:
City Cost Data
13 21 22 22 24 25 26 26 26 26 30 32 33 34 34 35 35 35 35 36 37 37 39 39 39 40 41 41 41 42 43 44 45 46 50 50 51 51 53 53 53 55 57 61 62 62 62 66 68 75
Suburban Cost Data
1- If the midpoints are 10, 15, 20, 25 and 30. The last class boundary of the distribution is: a) 25----30 b) 27.5----32.5 c) 20----35 d) 30----35
2- Mean of 10 items is 50 and S.D is 14. Find ?
a) 53920 b) 26960 c) 2696 d) 27453
3- The mean of an examination is 67, the median is 68, the mode is 69, The shape of the distribution is:
