Statistics Study notes of Engineering Mathematics for GATE CSE

 Example 1:

The measurements of the heights of all creatures in the world are data. All numerical characteristics are called variables. A large number of observations on a single variable can be summarized in a table of frequencies. Any particular pattern of variation followed is termed as distribution.

Frequency is that the number of times a specific value occurs during a set of knowledge. Usually, we might record the frequency of knowledge during a frequency table.

In statistics, mode, median, mean and range are values that represent a pool of numerical analysis. Thus, they are calculated from the pool of observations.

Measures of Central Tendency

Mode:

Mode is the most common value among the given observations. For example, a person selling ice creams might want to know that which flavour is the most popular.

Median:

Median is the middle value, dividing the number of data/values into 2 halves/parts. In other words, 50% of the observations are below the median and 50% of the observations are above the median. If the no of observations is odd, then the median is  observation. If the number of observations is even, then the median is the mean of  observations.

Mean:

Mean is the average of all the values in the set. Its value is given by . For example, an educator might want to understand the typical marks of a test in his class.

Example 2:

Find the mean, mode and median of the following set of points:

15, 14, 10, 8, 12, 8, 16, 13

Solution:

First arrange the point values in an ascending order (or descending order).

8, 8, 10, 12, 13, 14, 15, 16

Mean = (8+8+10+12+13+14+15+16)/8 = 96/8

= 12.

Mode = 8 (since it has maximum frequency)

The number of point values is 8, an even number.  Hence the median is that the average of the two middle values.

Median 

Skewness:

In a normal distribution, the mean, median, and mode are all the same values.  In various other symmetrical distributions it’s possible for the mean and median to be equivalent albeit there could also be several modes, none of which is at the mean. By contrast, in asymmetrical distributions, the mean and median aren’t equivalent. Such distributions are said to be skewed, i.e., quite half the cases are either above or below the mean.

Example 3:

The distribution of salaries shown in Figure has a pronounced skew distribution.

Figure: A distribution with a really large positive skew

The table shows the measures of central tendency for these data. The large skew leads to very different values for these measures. No single measure of central tendency is sufficient for data like these. There is no need to summarize a distribution with a single number. When the various measures differ, our opinion is that you should report the mean, median, and either the tri-mean or the mean trimmed 50%. Sometimes it is really worth reporting the mode as well. In the median, the median is generally reported to summarize the centre of skewed distributions. You will hear about median salaries and median prices of houses sold, etc and many more. This is better than reporting only the mean, but it would be informative and elaborative to hear more statistics.

Table: Measures of central tendency

Measures of Dispersion:

Dispersion measures the degree of scatteredness of the variable about a central value. The following are the measure of dispersion:

Range:

The range is the difference between the maximum and minimum values in the set. It is the simplest measure of variation to find.

RANGE = MAXIMUM VALUE – MINIMUM VALUE

Example 4:

Ten students were given a mathematics test. Time taken by them to complete the test is listed below. Find the range of these times

8 12 7 11 12 9 8 10 8 13 (in min.)

Solution:

It can be seen that the maximum time taken by a student to complete the test is 13 min and the minimum time taken is 7 min. So,

Range = Max. value – Min. value = 13 – 7 = 6 min

Mean Deviation:

It is the arithmetic mean of the absolute deviations of the terms of the distribution from its statistical mean. It is the least about the median.

Mean Deviation for Ungrouped Data:

Let x₁, x₂, x₃ ….x_n are n values of variable X and k be the statistical mean (A.M, median, mode) about which we have to find the mean deviation. The mean deviation about k is given by

Mean Deviation for Grouped data

Discrete Frequency Distribution:

where d_i = x_i – k, f_i be frequencies and N = total frequency.

The mean of given discrete frequency distribution is given by

The median of given discrete frequency distribution is found out by arranging the observations in ascending order and then calculating cumulative frequency. The observation, whose cumulative frequency is equal to or just greater than N/2, is the required median.

Continuous Frequency Distribution:

The mean of a continuous frequency distribution is calculated with the assumption that the frequency in each class is centred at its mid-point.

 where di=xi-k, fi be frequencies and N=total frequency

Arithmetic mean:

a = assumed mean, h = common factor and N = total frequency.

where,

l = lower limit of median class

f = frequency of the median class

h = width of the median class

c = cumulative frequency of the class just preceding the median class.

Example 5:

50 villages are inspected for calculating the total number of towers. The list below shows the distribution in numbers. Find the mean deviation of the distribution.

No. of towers 5, 6, 7, 8, 9, 10

No. of villages 8, 12, 9, 15, 4, 2

Solution:

Mean can be calculated from the formula given above as

Mean distribution is calculated as

Variance:

The variance of a variate is the arithmetic mean of the squares of all deviations from the mean.

Standard Deviation:

It is the proper measure of dispersion about the mean of a set of observations and is expressed as the positive square root of variance.

Example 6:

In the previous example, we have calculated the mean deviation of the data.

Using the same data calculate variance and standard deviation.

Solution:

Variance is given as:

Now, Standard Deviation is

Analysis of Frequency Distribution (Measures of Variability)

In order to compare the variability of two series with the same mean, which is measured in different units, merely calculating the measures of dispersion are not sufficient, but we require such measures which are independent of the units. The measure of variability which is independent of units is called the coefficient of variation (C.V.) and is defined as

Example 7:

From the data in the above two examples, find the coefficient of variation (C.V.) of no. of towers.

Solution:

From the formula above C.V. can be calculated as

C.V. = 0.6145 (which is the S.D.)/7.02 (which is the mean) × 100 = 8.7535

Example 8:

Given sets

A = {0,5,10,15,25,30,40,45,50,71,72,73,74,75,76,77,78,100} and

B = {0,22,23,24,25,26,27,28,29,50,55,60,65,70,75,80,85,90,95,100}.

Here simple inspection will indicate a very different distribution, however, it is found that

Min. = 0, Max. = 100

So range = 100,

Median = 50,

Middle of bottom half of the set (Q1) = 25,

Middle of the upper half of the set (Q3) = 75 is the same in both cases.

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