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Statistics Formula helps with the process of collection, classifying, arranging, interpreting and presenting, organizing a given volume of data. Data can be represented in various formats and is thus measured using different methods.
- Data is collected using both qualitative and quantitative methods.
- Measures of central value are known as the various methods of calculating the central values of a given data.
- The measure of Variation gives information about the degree to which individual data is clustered or deviates from the average value in a distribution.
- Statistics is simply a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data and datasets.
- It provides methods for summarizing and describing data, drawing conclusions from data, and making decisions based on data.
Read More: Measures of Dispersion
| Table of Content |
Key Terms: Statistics, Measures of Central Value, Measures of Variability, Mean, Median, Mode, Variance, Standard Deviation, Measures of Dispersion, Probability
What is Statistics?
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Statistics is widely associated with classifying, collecting, arranging and presenting numerical data.
- It enables interpreting a variety of results as well as forecasting several possibilities.
- Statistics is known to deal with facts, observations and information in the form of numeric data alone.
- Statistics gives us the ability to find various measures of central tendencies and the deviation of various values from the center.
It is further classified into several forms, including:
- Mean
- Median
- Mode
- Variance
- Standard Deviation
Mean
The most common central value is the Mean. The mean of a group of observations is the value which is equally shared among all observations. The arithmetic mean is the sum total of all observations divided by the number of items.
Median
Median of a group of observations is the value of the variable which divides the group into two equal parts. In other words, the median is the value which exceeds and is exceeded by the same number of observations.
Mode
Mode of a set of observations is the value of the observations that occur the most number of times.
Variance
Variance is a measure of statistics showing how a particular data differentiates from the mean. Variance is the difference of deviation from the actual value.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion of a set of values from its mean or average. It is calculated by finding the square root of the variance, which is the average of the squared differences between each value and the mean. In other words, it measures how much the data deviates from the average value.
The formula for calculating the standard deviation is:
| σ = √(Σ(xi - μ)² / N) |
Herein,
- σ is the standard deviation
- xi is each value in the dataset
- μ is the mean of the dataset
- N is the number of values in the dataset
Mean, Media, and Mode Video Explanation
Also Read:
Statistics Formula
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Formulae of different Statistics methods are:
Mean
Mean =Sum of observations/Total number of observations
Median
If n is odd, then the following formula is used: Value of (n+1/2th observation.
If n is even, then the following formulae is used: {Value of (n/2) th observation + (n/2 +1)th observation/2}
Mode
Most number of times occurring value in the data.
Mode= 3 Median – 2 Mean
Standard Deviation = Square root of Variance
Also, read: Difference Between Mean, Median, Mode
List of Statistics Formulas
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The statistics formula list can be represented as:
| Terms | Formula Used | Observations/Examples |
|---|---|---|
| Mean | \(\bar{x}=\frac{\sum x}{n}\) | x = Given Observations, n = Total No. of Observations |
| Median | In case n is odd, M = \(\begin{array}{l}(\frac{n+1}{2})\end{array}\) term; In case n is even, M = \(\begin{array}{l}\frac{(\frac{n}{2})^{th}term+(\frac{n}{2}+1)^{th}term}{2}\end{array}\) | n = Total Number of Observations |
| Mode | Value that occurs most frequently | 3, 2, 1, 3, 3, 4, 3, 1, 1, 2, 2, 3, 3 (Thus, 3 occurs the most in the dataset |
| Standard Deviation | \(\begin{array}{l}S = \sigma = \sqrt{\frac{\sum (x-\bar{x})^{2}}{n}}\end{array}\) | x = Given Observations, \(\bar{x}\) = Mean, n = Total No. of Observations |
| Variance | \(\begin{array}{l}\frac{\sum (x-\bar{x})^{2}}{n}\end{array}\) | x = Given Observations, \(\bar{x}\) = Mean, n = Total No. of Observations |
Things to Remember
- The arithmetic mean is the sum total of all observations divided by the number of items and is used to calculate the average.
- Median is the value which exceeds and is exceeded by the same number of observations.
- Mode of a set of observations is the value of the observations that occurs the most number of times
- Mode= 3 Median – 2 Mean
- Variance is a measure of statistics showing how a particular data differentiates from the mean. Variance is the difference of deviation from the actual value
- Standard of Deviation is the square root of Variance.
Read More:
Previous Year Questions
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Sample Questions
Ques. A batsman scored the following numbers of runs in six innings.
36, 35, 50, 46, 60, 55. Calculate the mean runs scored. (1 Mark)
Ans. Mean runs scored = 36+35+50+46+60+55/6 = 282/6 = 47
Ques. Find the mean of the first 6 natural numbers. (1 Mark)
Ans. The mean of the first 6 natural numbers ARE 1, 2, 3, 4, 5, 6.
Mean= 21/6 = 3.5
Ques. Find the median of the data 23, 33, 48, 13, 15, 26, 37. (2 Marks)
Ans. Arrange the data in ascending order we get,
13, 15, 23, 26, 33, 37, 48
Here, n = 7 which is odd
Median = Value of (7+1/2) th observation = Value of 4 th observation = 26.
Ques. Find the median of the data 38, 32, 43, 44, 47, 26, 40, 46, 33. (2 Marks)
Ans. Arrange the data in ascending order we get,
26, 32, 33, 38, 40, 43, 44, 46, 47
Here, n = 9 which is odd
Median = Value of (9+1/2) th observation = Value of 5 th observation = 40.
Ques. Find Mode of the data if it is given that Mean = 3 and Median = 1. (2 Marks)
Ans. Mode = 3 Median – 2 Mean.
= 9- 2 = 7
Thus the Mode is 7.
Ques. Find Mode of the data 1, 1, 2,4,2,1,2,2,4. (2 Marks)
Ans. Arrange the numbers with same values together,
1,1,1,2,2,2,2,3,4,4
Clearly 2 occurs maximum numbers of time so 2 is the mode of the given data.
Ques. The median of the observations 11, 12, 14, 17, x+2, x+4, 31, 32, 35, 41, arranged in ascending order is 24. Find the value of x. (3 Marks)
Ans. Here, n = 10 which is even.
If n is even, then the following formulae is used
Median: {Value of (n/2) th observation + (n/2 +1)th observation/2}
24= 5th observation + 6th observation/2
24 = (x+2) +(x+4)/2
24= 2x+6/2
24 = x+3
X= 21
The value of x is 21.
Ques. Find the median of the data: 20, 26, 60, 49, 36, 32, 31, 33, 52. If 26 is replaced by 53, what will be the new median? (3 Marks)
Ans. Arrange the data in ascending order we get
20, 26,31,32,33,36,49,52, 60
Here, n = 9 which is odd
Median = Value of (9+1/2) th observation = Value of 5th observation = 33
Hence the median is 33.
When 26 is replaced by 53 the data in ascending order is:
20, 31,32,33,36,49,52,53, 60
Median = Value of (9+1/2) th observation = Value of 5th observation = 36
Ques. What is the mode of the set S = 1, 3, 3, 6, 8, 9, 5, 3, 4, 4, 5, 4, 4, 4, 4, 8? (2 Marks)
Ans. Since mode is the value that appears most frequently in a dataset, it can be said:
S = 1, 3, 3, 6, 8, 9, 5, 3, 4, 4, 5, 4, 4, 4, 4, 8
It can be seen that 4 occurs the most. Thus, 4 is the mode of the dataset.
Ques. The minimum steps climbed by a person everyday during week were measured as:
Determine the mean of the steps climbed. (3 Marks)
Ans. The number of steps the person has climbed in a week = 35, 30, 27, 32, 23, 28.
Thus, we can say:
Mean = sum of observation / total no of observations (Here, observations = steps)
= (35 + 30 + 27 + 32 + 23 + 28) / 6
= 175/6
= 29.17
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