Introduction to Statistics
Statistics is a branch of Mathematics. It deals with collecting, organizing, analyzing, interpreting, and then presenting it to reach some conclusion. It is similar to the job an accountant does, except that the accountant just handles financial data, while a statistician handles all kinds of data.
A statistician makes use of quantitative tools for gathering and evaluating big chunks of data. These days all businesses are data businesses, so the value of a statistician is immense because he can read data and bring our meaning and draw conclusions out of it. He uses statistics to help businesses make efficient and well-informed decisions. The use of statistics is not limited to any field or genre – where there is data, there are statistics.
In this age of planning, Statistics is an indispensable tool. Countries across the globe use statistical data and techniques of statistical analysis for economic development and in solving problems like wages, price, time series analysis, demand analysis.
Businesses that use statistics have an edge over their competitors through better planning and accurate assumptions and predictions. Through statistical tools that predict cyclical and general economic fluctuations, they handle future uncertainties better. It is an irreplaceable tool of production control. Statistical tools are also becoming the go-to trick for business executives to read their customers and better meet their demands.
Equality control uses industry statistics. Various statistical tools like inspection plans and control charts are popularly used to find out the aptness of a product in production engineering.
Statistics become a golden hen in the hands of creative analysts. Therefore, it has gained traction in diverse fields and is being used by individuals, like insurance companies, social workers, labor unions, trade associations, chambers, and politicians.
The applications of statistics are so diverse that with the right statistical tool and mathematical and statistical knowledge, major social and economic problems are being solved. This is the reason statisticians are found in agriculture, business, industry, computer science, medical sciences – you name it!
Statistics is divided into two branches – Inferential Statistics and Descriptive Statistics. As the name implies, the first branch aims at making predictions or generalizations through analysis while the second branch aims to describe and grade the visible traits of data.
The term measures of central tendency are represented as a single value that is used to define a collection of data by arranging the central position within that set of data. It is also categorized as a statistical summary. It helps to make the statistical summary of large organized data. The most common method of measure of central tendency in statistics is Arithmetic Mean.
For Example: Have you ever observed the daily temperature records while reading the newspaper early in the morning. As we know temperature varies throughout the day, yet how can a single temperature represent the weather condition of the whole day.? Or when students get a scorecard in the examination, the best way to calculate the students’ performance is to calculate the aggregate percentage of all the subjects.
The representation of large amounts of data in a single value makes it easy to understand and analyze the collection of data or to get the required information out of it. Let us now understand the arithmetic mean in statistics, how to find the arithmetic mean in statistics, arithmetic mean examples, arithmetic formula etc.
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What is Arithmetic Mean in Statistics ?
Arithmetic Mean is the most common measurement of central tendency. According to the layman, the mean of data represents an average of the given collection of the data. It is equivalent to the sum of all the observations of a given data divided by the total number of observations.
The mean of data for n values in a set of data namely\[p_{1}, p_{2}, p_{3} ----- p{_n}\] is given by:
\[\overline{p} = \frac{p_{1}, p_{2}, p_{3} ----- p{_n}}{n}\]
For calculating the arithmetic when the number of observations along with the frequency of observation is given such that \[p_{1},p_{2}, p_{3} ---p_{n}\] are the recorded observation and \[f_{1},f_{2}, f_{3} ---f_{n}\] are the corresponding frequencies of the observation the,
\[\overline{x} = \frac{f_{1}p_{1}, f_{2}p_{2}, f_{3}p_{3} ------ f_{n}p_{n}}{f_{1},f_{2}, f_{3} ---f_{n}}\]
The above Arithmetic mean formula is expressed as
\[\frac{\sum f_{i}x_{i}}{\sum f_{i}} \]
The above-given arithmetic mean formula is used to calculate the mean when the data given is ungrouped. For calculating the mean of group data, we calculate the class marks.
Midpoints of the classmark is computed as:
\[\text{Midpoint} = \frac{\text{Upper Limit + Lower Limit}}{2}\]
The method discussed above is the calculation of the arithmetic mean by the direct method.
Arithmetic Mean Formulas in Statistics
Arithmetic mean formula in statistics for grouped data
\[\overline{x} = \frac{\sum f_ix_i}{\sum f_i} \]
Here,
\[\overline{x}\] = Arithmetic mean
f = Frequency
X = variable
\[\sum f\] = Sum of frequencies
Arithmetic Mean Formula in Statistics for ungrouped Data
\[\text{Midpoint of class Interval} = \frac{\text{Upper Limit + Lower Limit}}{2}\]
The arithmetic mean of a collection of numbers (from x1to xn) is derived by the formula
\[x = \frac{1}{n}\]
\[\sum_{i=1}^{n} x_{i} = \frac{x_{1} + x_{2} + x_{3} ------ + x_{n}}{n}\]
How to calculate Arithmetic Mean in Statistics?
There are two steps to find the arithmetic mean in statistics: sum up all the numbers given in a set and then divide it by the total number of items in your set. The arithmetic mean is found similarly to a sample mean.
For example: Calculate the arithmetic mean for the average driving speed for one bus over a 5hours journey. 50 mph , 23mph, 60mph, 65mph, 30mph
Step 1: Addition all the numbers in a data set : 52 + 23 + 60 + 65 + 30
Step 2: Divide by the total number of items in a given data set. There are 5 numbers in an above set
Accordingly,
= \[\frac{52+ 23 + 60 + 65 + 30}{5}\]
= \[\frac{230}{5}\]
= 46
Hence, the average driving speed of a bus is 46 mph.
Arithmetic Mean Example in Statistics
Let us understand the concept of arithmetic mean clearly through an example:
1. In a class of 30 Students, Marks Obtained by the Students in Science out of 50 are Given Below in Tabular Form. Calculate the Arithmetic Mean of the Data.
Solution:
\[\text{Midpoint formula} = \frac{\text{Upper Limit + Lower Limit}}{2}\]
The arithmetic mean of above data is
\[\overline{x} = \frac{\sum f_{i} x_{i}}{\sum f_{i}} = \frac{1020}{30} = 34\]
What is meant in Statistics with Example?
Mean is simply the average of the given set of values in a data set. The mean is represented by \[\overline{x}\].
Mean = Sum of the given values in a data set/ Total number of values
Generally, the mean is defined for the average of the sample, whereas the average denotes the addition of all the values to the total number of values. Logically, average and mean both are similar terms.
For example: Calculate the mean of the given values: 5,6,3,2,1,8
Mean = \[\frac{( 5 + 6+ 3+ 2+ 1+ 8)}{5}\]
= \[\frac{25}{5}\]
= 5
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Solved Example
1. Calculate the Arithmetic Mean from the Following Data
Solution:
Midpoint point for \[1^{st}\] class interval = \[\frac{15 + 18}{2} = 16.5 \]
\[\overline{x} = \frac{\sum f_{m}}{\sum f_{i}}\]
=\[\frac{ 2175}{100}\]
= Rs. 21.75
2. If the arithmetic mean of the 14 different observations are 26, 12, 15, x, 17, 9, 11, 18, 16, 28, 20, 22, 8, is 17. Find the missing observation.
Solution: 14 observations are = 26,12,15,x,17,9,11,18,16,28,20,22,8,
Arithmetic mean = 17
We know that,
\[\text{Arithmetic Mean} = \frac{\text{Sum of total observations}}{\text{Total number of observations}}\]
\[17 = \frac{(216 + x)}{14}\]
17 x 14 = 216 + x
x = 238 - 216
x= 22
Hence, missing observation is 22.
3. The marks obtained by 7 students in science class tests are 20, 22, 24, 26, 28, 30, 10. Find the arithmetic mean.
Solution:
\[\overline{x} = \frac{20 + 22 + 24 + 26 + 28 + 30 + 10}{7} = 22.8 \]
Hence, the arithmetic mean of 7 students is 22.8.
Quiz Time
1. What should be the value of 'a', if the arithmetic mean between a and 10 is 30
45
60
50
53
2. The series obtained by adding the term of an arithmetic sequence is known as,
Arithmetic Series
Harmonic Series
Geometric Series
Infinite Series
3. What will the arithmetic mean between 1 + x + x and 1 - x + x ?
1 - x
1 + x
2 - x
2 + x
Fun Facts
Tycho Brahe was the first to use the concept of the arithmetic mean.
An Indian Mathematician and astronomer Brahmagupta is the father of the arithmetic mean.
The word arithmetic is derived from the Greek noun arithmos meaning "number".
FAQs on How to Find Arithmetic Mean in Statistic
1. What are the merits and demerits of arithmetic mean?
Merits of Arithmetic Means
The calculation of arithmetic mean is easy as it requires basic knowledge of Maths such as addition, subtraction, multiplication, and division of numbers.
The meaning of the arithmetic mean can be easily understood such as the value or unit or cost per unit etc.
The value of the arithmetic mean is always definite as it is defined rigidly.
It can be widely used in advanced statistical analysis as it has competency for further algebraic operation.
A comparison of the data of two or more groups can be easily done through arithmetic mean.
All the values of data are taken into consideration while calculating arithmetic mean .
Demerits of Arithmetic Mean
The calculation of arithmetic mean is not possible if all the items of the series are not available.
The calculation of the arithmetic mean cannot be done just by observing the series such as median or mode.
Arithmetic means cannot be represented on graph paper.
It is not possible to calculate the arithmetic mean for qualitative data such as intelligence, honesty, victory etc.
It is not possible to compute arithmetic mean in case of open- end class distribution as it cannot be calculated without making assumptions about the class size.
Sometimes arithmetic means giving illogical results. For example- if teachers say the average number of girls in a class is 28.97, it sounds illogical.
2. What are some basic properties of arithmetic mean?
Here are some of the basic arithmetic mean properties:
If X̅ is considered as the mean of n observations x1, x2 ……xn , then the mean of x1-a, x2-a …xn– a is X̅ – a, where a is any real number.
If X̅ is considered as the mean of n observations x1, x2 ……xn , then the mean of x1/a, x2/a …xn/a is X̅ / a, where a is any non-zero.
If X̅ is considered as the mean of x1, x2 ……xn , then the mean of ax1, ax2…axn is an X̅ where a is any number that is different from zero. If every observation is multiplied by a non-zero a. Then the mean will also be multiplied by a.
If X̅ is considered as the mean of x1, x2 ……xn , then prove that Σni=1( xi - X̅) is equal to zero.
3. When do we use Inferential Statistics?
Inferential statistics are used to compare differences between treatment groups. It is very much like a science experiment. An example can be a test to see if sugar affects the behaviour of mice. One group of mice gets sugar and another group of mice gets a placebo. These two groups would be called the treatment groups. The mice are the sample subjects and are used to make generalizations about the larger population of subjects – mice. Generally, inferential statistics works on the principle that a test-statistic value is calculated based on a particular formula. There is an assumption that is proven through inference, just like a science experiment has a hypothesis that is proven or disproven through the experiment. For inferential statistics to reach a solid conclusion, sample size matters. The larger the sample size, the stronger the statistic will be.
4. When do we use Descriptive Statistics?
An example of using descriptive statistics would be about the average score of students in a maths test. It does not draw conclusions or inferences, it just presents a large amount of data in a meaningful manner and the process reduces big data into a succinct summary.
5. Is statistics used on the internet?
All businesses use statistics to find out about their customers and design their products to better satisfy their customers and hence do better business. Statistics use data and as more people use the internet, more people provide data. This data is collected by websites and social media, which uses it in multiple ways to boost their business. So much so that now ethical use of data collection is a major issue and policies are being made to safeguard the user’s privacy and personal data.
