GIS Statistics (Measures of Central Tendency, Measures of Deviation and Variance) (Especially for GATE-Geospatial 2022)

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Statistics

Statistics is a mathematical science relating to the collection, analysis, interpretation, and presentation of data. Many models built using GIS are statistical in natural and many statistical techniques are used for analysis of GIS data. Statistics is also important in understanding issues of error and uncertainty in GIS data.

GIS statistics represents differences:

This Diagram Shows GIS Statistics Represents Differences

Mean

The mean () of a set of values is its statistical average, such that, if represents a set of k values

Or

For example, we have a book having 20 chapters, and the numbers of pages per chapter sequentially are

Therefore, the mean of pages per chapter

This Diagram Shows Frequency and Normal Distribution

The mean of data file values with a normal distribution is the data file value associated with the peak of the curve – the point where the distribution balances.

Bell curve is a common type of graph that has more or less the shape of a bell.

Median

Median is also one type of statistical average. Median is merely the ‘middle value’ of a list of numbers, after we have arranged data in order of magnitude.

For example, we have a book having 20 chapters, and number of pages per chapter sequentially is

Now, we sort the numbers from the smallest to the largest

There is an even number of values, so the middle is the sum of both the middle numbers divided by two. In this case it is

It there is an odd number of the values, then the middle number is the median. For example, in the following case:

The median is 5.

Mode

In statistics, the mode is the value that has the largest number of observations; namely, the most frequent value or values. The easiest way to look at modes is on histograms.

For example, we have a book having 20 chapters, and number of pages per chapter sequentially is

The histogram of these numbers is shown figure.

This Diagram Shows Histogram

Now, let us consider the value that repeats most often. It looks like this highest peak on our histogram. There are three values that appear most often; 3,4, and 6, so all these values are modes.

Modes are often used for so-called qualitative data, that is, data that describes qualities rather than quantities. If there are several modes data is called bimodal, trimodal, or multimodal, accordingly.

In the following example:

mode is 85.

Variance

The mean of a set of values locates only the average value – it does not adequately describe the set of values by itself. It is helpful to know much the data varies from its mean. However, a simple average of the difference between each value and the mean equals zero in every case, of the mean.

Therefore, the square of these differences are averaged so that a meaningful number results. Variance uses the following formula:

Where

a particular pixel

the number of pixels

set of values

mean of all values

Standard Deviation

Since the variance is expressed in units squared, a more useful value is the square root of the variance, which is expressed in units and can be related back to the original values. The square root of the variance is the standard deviation. The standard deviation for a set of values Q is computed as follows:

Covariance

Covariance measures the tendencies of data file values in the same pixel, but in different bands, to vary with each other, in relation to the means of their respective bands. Theoretically, whereas variance is the average square of the differences between values and their mean in one band, covariance is the average product of the differences of corresponding value sin two different bands from their respective means.

The sample covariance is computed with the following equation:

Where

a particular pixel

the number of pixels

data file values in two bands

mean of data file values at band Q

mean of data file values at band R

Like variance, covariance is expressed in units squared.

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