Statistics is a branch of mathematics that deals with the study of collecting, analyzing, interpreting, presenting, and organizing data in a particular manner. Statistics is defined as the process of collection of data, classifying data, representing the data for easy interpretation, and further analysis of data. Statistics also is referred to as arriving at conclusions from the sample data that is collected using surveys or experiments. Different sectors such as psychology, sociology, geology, probability, and so on also use statistics to function.
Statistics is used mainly to gain an understanding of the data and focus on various applications. Statistics is the process of collecting data, evaluating data, and summarizing it into a mathematical form. Initially, statistics were related to the science of the state where it was used in the collection and analysis of facts and data about a country such as its economy, population, etc. Mathematical statistics applies mathematical techniques like linear algebra, differential equations, mathematical analysis, and theories of probability.
There are two methods of analyzing data in mathematical statistics that are used on a large scale:
Descriptive Statistics
The descriptive method of statistics is used to describe the data collected and summarize the data and its properties using the measures of central tendencies and the measures of dispersion.
Inferential Statistics
This method of statistics is used to draw conclusions from the data. Inferential statistics requires statistical tests performed on samples, and it draws conclusions by identifying the differences between the 2 groups. Tests calculate the pvalue that is compared with the probability of chance(α) = 0.05. If the pvalue is less than α, then it is concluded that the pvalue is statistically significant.
The collection of observations and facts is known a data. These observations and facts can be in the form of numbers, measurements, or statements. There are two different kinds of data i.e. Qualitative data and quantitative data. Qualitative data is when the data is descriptive or categorical and quantitative data is when the data is numerical information.
Once we know the data collection methods, we aim at representing the collected data in different forms of graphs such as a bar graph, line graph, pie chart, stem and leaf plots, scatter plot, and so on. Before the analysis of data, the outliers are removed that are due to the invariability in the measurements of data. Let us look at different kinds of data representation in statistics.
Statistics being a broad term used in various forms, different models of statistics are used in different forms. Listed below are a few models:
The measure of central tendency and the measure of dispersion are considered as the basis of descriptive statistics. The representative value for the given data is the measure of central tendency that gives us an idea of where data points are centered. This is done to find how the data are scattered around this centered measure. We use mean, median, and mode to find the central measures of tendency. In our daytoday life, we find the average height of the students, the average income, the average score in exams, or of the player. The different measures of central tendency for the data are:
Mean, Median and Mode in Statistics
Mean is considered the arithmetic average of a data set that is found by adding the numbers in a set and dividing by the number of observations in the data set. The middle number in the data set while listed in either ascending or descending order is the median. Lastly, the number that occurs the most in a data set and ranges between the highest and lowest value is the mode. For n number of observations, we have
The measures of central tendency do not suffice to describe the complete information about a given data. Thus we need to describe the variability by a value called the measure of dispersion. The different measures of dispersion are:
Mean Deviation For ungrouped data
In statistics, the frequency distributions of data can be discrete data or continuous. For n number of individual observations
x_{1}, x_{2}, x_{3}, x_{r}, . . . . . x n x_{1},x_{2},x_{3},x_{r},.....x_{n}, the mean deviation about mean and median are calculated as follows:
Mean Deviation for ungrouped data = sum of deviation/number of observations =
The measurements of the data units are clearly shown in such a frequency distribution. Let there be n distinct data points x_{1}, x_{2}, x_{3}, x_{r} , . . . . . x_{n}, occurring with frequencies f_{1 }, f_{2} , f_{3} . . . . f_{n}.
a) Mean deviation about mean
b) Mean deviation about median
Here the data points take any value within a range and they are continuous. They can be measured and represented by using intervals on the real number line. The frequency in which data are arranged in classes is not countable.
a) Mean deviation about mean
The mean of the continuous frequency distribution is centered at its midpoint in each class. Then the same procedure is followed as in the case of discrete frequency distribution.
b) Mean deviation about median
Median = where the median class is the class interval whose cf is ≥ N/2, N the sum of frequencies, l, f, h, and C are, the lower limit, the frequency, the width of the median class and C the cumulative frequency of the class just preceding the median class. After finding the median,  x_{i}  M is obtained.
We have the other prominent methods in statistics to find the proper measure of dispersion, known as the variance and the standard deviation. While finding the mean deviation about the mean and the median, there arises a difficulty in taking squares of all the deviations.
Coefficient of Variation
We compare the coefficient of variations of two or more frequency distributions. This coefficient of variation in statistics is the ratio of the standard deviation to the mean, expressed in percentage.
The distribution that has a greater coefficient of variation has more variability around the central value than the distribution having a smaller value of the coefficient of variation.
Important Notes
406 videos217 docs164 tests


Explore Courses for Class 10 exam
