Basic Statistics and Data Analysis

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Tagged: Measure of Dispersion

Basic Statistics / Measure of Dispersion

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Standard Deviation: A Measure of Dispersion

Standard Deviation

The standard deviation is a widely used concept in statistics and it tells how much variation (spread or dispersion) is in the data set. It can be defined as the positive square root of the mean (average) of the squared deviations of the values from their mean.
To calculate the standard deviation one have to follow these steps:

  1. First, find the mean of the data.
  2. Take the difference of each data point from the mean of the given data set (which is computed in step 1). Note that, the sum of these differences must be equal to zero or near to zero due to rounding of numbers.
  3. Now computed the square the differences that were obtained in step 2, It would be greater than zero, that it, I will be a positive quantity.
  4. Now add up all the squared quantities obtained in step 3. We call it the sum of squares of differences.
  5. Divide this sum of squares of differences (obtained in step 4) by the total number of observation (available in data) if we have to calculate population standard deviation ($\sigma$). If you want t to compute sample standard deviation ($S$) then divide the sum of squares of differences (obtained in step 4) by the total number of observation minus one ($n-1$) i.e. the degree of freedom. Note $n$ is the number of observations available in the data set.
  6. Find the square root (also known as under root) of the quantity obtained in step 5. The resultant quantity in this way known as the standard deviation for the given data set.

The sample standard deviation of a set of $n$ observation, $$X_1, X_2, \cdots, X_n$$ denoted by $S$ is
\begin{aligned}
\sigma &=\sqrt{\frac{\sum_{i=1}^n (X_i-\overline{X})^2}{n}}; Population\, Standard\, Deviation\\
S&=\sqrt{ \frac{\sum_{i=1}^n (X_i-\overline{X})^2}{n-1}}; Sample\, Standard\, Deviation
\end{aligned}
The standard deviation can be computed from variance too as $S= \sqrt{Variance}$.

The real meaning of the standard deviation is that for a given data set 68% of the data values will lie within the range $\overline{X} \pm \sigma$ i.e. within one standard deviation from mean or simply within one $\sigma$. Similarly, 95% of the data values will lie within the range $\overline{X} \pm 2 \sigma$ and 99% within $\overline{X} \pm 3 \sigma$.

Examples of Standard Deviation and Variance

A large value of standard deviation indicates more spread in the data set which can be interpreted as the inconsistent behaviour of the data collected. It means that the data points tend to away from the mean value. For the case of smaller standard deviation, data points tend to be close (very close) to mean indicating the consistent behaviour of data set.
The standard deviation and variance both are used to measure the risk of a particular investment in finance. The mean of 15% and standard deviation of 2% indicates that it is expected to earn a 15% return on investment and we have 68% chance that the return will actually be between 13% and 17%. Similarly, there are 95% chance that the return on the investment will yield an 11% to 19% return.

Basic Statistics / Measure of Dispersion

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Range: Measure of Dispersion

Measure of Central Tendency provides typical value about the data set, but it does not tell the actual story about data i.e. mean, median and mode are enough to get summary information, though we know about the center of the data. In other words, we can measure the center of the data by looking at averages (mean, median, mode). These measure tell nothing about the spread of data. So for more information about data we need some other measure, such as measure of dispersion or spread.

Spread of data can be measured by calculating the range of data; range tell us over how many numbers of data extends. Range (an absolute measure of dispersion) can be found by subtracting highest value (called upper bound) in data from smallest value (called lower bound) in data. i.e.

Range = Upper Bound – Lowest Bound
OR
Range = Largest Value – Smallest Value

This absolute measure of dispersion have disadvantages as range only describes the width of the data set (i.e. only spread out) measure in same unit as data, but it does not gives the real picture of how data is distributed. If data has outliers, using range to describe the spread of that can be very misleading as range is sensitive to outliers. So we need to be careful in using range as it does not give the full picture of what’s going between the highest and lowest value. It might give misleading picture of the spread of the data because it is based only on the two extreme values. It is therefore an unsatisfactory measure of dispersion.

However range is widely used in statistical process control such as control charts of manufactured products, daily temperature, stock prices etc., applications as it is very easy to calculate. It is an absolute measure of dispersion, its relatives measure known as the coefficient of dispersion defined the the relation

\[Coefficient\,\, of\,\, Dispersion = \frac{x_m-x_0}{x_m-x_0}\]

Coefficient of dispersion is a pure dimensionless and is used for comparison purpose.

Basic Statistics / Measure of Dispersion

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Absolute Measure of Dispersion

Absolute Measure of Dispersion gives an idea about the amount of dispersion/ spread in a set of observations. These quantities measure the dispersion in the same units as the units of original data. Absolute measures cannot be used to compare the variation of two or more series/ data sets. A measure of absolute dispersion does not in itself, tell whether the variation is large or small.

Commonly used Absolute Measure of Dispersion are:

  1. Range
  2. Quartile Deviation
  3. Mean Deviation
  4. Variance or Standard Deviation

The details about these Absolute measures of Dispersion or spread are:

Range

The Range is the difference between the largest value and the smallest value in the data set. For ungrouped data, let $X_0$ is the smallest value and $X_n$ is the largest  value in a data set then the range (R) is defined as
$R=X_n-X_0$.

For grouped data Range can be calculated in three different ways
R=Mid point of the highest class – Midpoint of the lowest class
R=Upper class limit of the highest class – Lower class limit of the lower class
R=Upper class boundary of the highest class – Lower class boundary of the lowest class

Quartile Deviation (Semi-Interquantile Range)

The Quartile deviation is defined as the difference between the third and first quartiles, and half of this range is called the semi-interquartile range (SIQD) or simply quartile deviation (QD). $QD=\frac{Q_3-Q_1}{2}$
The Quartile Deviation is superior to range as it is not affected by extremely large or small observations, anyhow it does not give any information about the position of observation lying outside the two quantities. It is not amenable to mathematical treatment and is greatly affected by sampling variability. Although Quartile Deviation is not widely used as a measure of dispersion, it is used in situations in which extreme observations are thought to be unrepresentative/ misleading. Quartile Deviation is not based on all observations therefore it is affected by extreme observations.

Note: The range “Median ± QD” contains approximately 50% of the data.

Mean Deviation (Average Deviation)

The Mean Deviation is defined as the arithmetic mean of the deviations measured either from the mean or from the median. All these deviations are counted as positive to avoid the difficulty arising from the property that the sum of deviations of observations from their mean is zero.
$MD=\frac{\sum|X-\overline{X}|}{n}\quad$ for ungrouped data for mean
$MD=\frac{\sum f|X-\overline{X}|}{\sum f}\quad$ for grouped data for mean
$MD=\frac{\sum|X-\tilde{X}|}{n}\quad$ for ungrouped data for median
$MD=\frac{\sum f|X-\tilde{X}|}{\sum f}\quad$ for grouped data for median
Mean Deviation can be calculated about other central tendencies but it is least when deviations are taken as the median.

The Mean Deviation gives more information than the range or the Quartile Deviation as it is based on all the observed values. The Mean Deviation does not give undue weight to occasional large deviations, so it should likely be used in situations where such deviations are likely to occur.

Variance and Standard Deviation

This absolute measure of dispersion is defined as the mean of the squares of deviations of all the observations from their mean. Traditionally for population variance is denoted by $\sigma^2$ (sigma square) and for sample data denoted by $S^2$ or $s^2$.
Symbolically
$\sigma^2=\frac{\sum(X_i-\mu)^2}{N}\quad$ Population Variance for ungrouped data
$S^2=\frac{\sum(X_i-\overline{X})^2}{n}\quad$ sample Variance for ungrouded data
$\sigma^2=\frac{\sum f(X_i-\mu)^2}{\sum f}\quad$ Population Variance for grouped data
$\sigma^2=\frac{\sum f (X_i-\overline{X})^2}{\sum f}\quad$ Sample Variance for grouped data

The variance is denoted by Var(X) for random variable X. The term variance was introduced by R. A. Fisher (1890-1982) in 1918. The variance is in square of units and the variance is a large number compared to observation themselves.
Note that there are alternative formulas to compute Variance or Standard Deviations.

The positive square root of the variance is called Standard Deviation (SD) to express the deviation in the same units as the original observation themselves. It is a measure of the average spread about the mean and is symbolically defined as
$\sigma^2=\sqrt{\frac{\sum(X_i-\mu)^2}{N}}\quad$ Population Standard for ungrouped data
$S^2=\sqrt{\frac{\sum(X_i-\overline{X})^2}{n}}\quad$ Sample Standard Deviation for ungrouped data
$\sigma^2=\sqrt{\frac{\sum f(X_i-\mu)^2}{\sum f}}\quad$ Population Standard Deviation for grouped data
$\sigma^2=\sqrt{\frac{\sum f (X_i-\overline{X})^2}{\sum f}}\quad$ Sample Standard Deviation for grouped data
Standard Deviation is the most useful measure of dispersion is credited with the name Standard Deviation by Karl Pearson (1857-1936).
In some text Sample, Standard Deviation is defined as $S^2=\frac{\sum (X_i-\overline{X})^2}{n-1}$ on the basis of the argument that knowledge of any $n-1$ deviations determines the remaining deviations as the sum of n deviations must be zero. In fact, this is an unbiased estimator of the population variance $\sigma^2$. The Standard Deviation has a definite mathematical measure, it utilizes all the observed values and is amenable to mathematical treatment but affected by extreme values.

References

Introduction to Multivariate Statistics / Measure of Central Tendency / Measure of Dispersion / Multivariate Statistics

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Descriptive Statistics Multivariate Data set

Much of the information contained in the data can be assessed by calculating certain summary numbers, known as descriptive statistics such as Arithmetic mean (a measure of location), an average of the squares of the distances of all of the numbers from the mean (variation/spread i.e. measure of spread or variation), etc. Here we will discuss descriptive statistics multivariate data set.

We shall rely most heavily on descriptive statistics which is a measure of location, variation, and linear association. For descriptive statistics multivariate data set, let us start with a measure of location, a measure of spread, sample covariance, and sample correlation coefficient.

Measure of Location

The arithmetic Average of $n$ measurements $(x_{11}, x_{21}, x_{31},x_{41})$ on the first variable (defined in Multivariate Analysis: An Introduction) is

Sample Mean = $\bar{x}=\frac{1}{n} \sum _{j=1}^{n}x_{j1} \mbox{ where } j =1, 2,3,\cdots , n $

The sample mean for $n$ measurements on each of the p variables (there will be p sample means)

$\bar{x}_{k} =\frac{1}{n} \sum _{j=1}^{n}x_{jk} \mbox{ where }  k  = 1, 2, \cdots , p$

Measure of Spread

Measure of spread (variance) for $n$ measurements on the first variable can be found as
$s_{1}^{2} =\frac{1}{n} \sum _{j=1}^{n}(x_{j1} -\bar{x}_{1} )^{2} $ where $\bar{x}_{1} $ is sample mean of the $x_{j}$’s for p variables.

Measure of spread (variance) for $n$ measurements on all variable can be found as

$s_{k}^{2} =\frac{1}{n} \sum _{j=1}^{n}(x_{jk} -\bar{x}_{k} )^{2}  \mbox{ where } k=1,2,\dots ,p \mbox{ and } j=1,2,\cdots ,p$

The Square Root of the sample variance is sample standard deviation i.e

$S_{l}^{2} =S_{kk} =\frac{1}{n} \sum _{j=1}^{n}(x_{jk} -\bar{x}_{k} )^{2}  \mbox{ where }  k=1,2,\cdots ,p$

Sample Covariance

Consider n pairs of measurement on each of Variable 1 and Variable 2
\[\left[\begin{array}{c} {x_{11} } \\ {x_{12} } \end{array}\right],\left[\begin{array}{c} {x_{21} } \\ {x_{22} } \end{array}\right],\cdots ,\left[\begin{array}{c} {x_{n1} } \\ {x_{n2} } \end{array}\right]\]
That is $x_{j1}$ and $x_{j2}$ are observed on the jth experimental item $(j=1,2,\cdots ,n)$. So a measure of linear association between the measurements of  $V_1$ and $V_2$ is provided by the sample covariance
\[s_{12} =\frac{1}{n} \sum _{j=1}^{n}(x_{j1} -\bar{x}_{1} )(x_{j2} -\bar{x}_{2}  )\]
(the average of product of the deviation from their respective means) therefore

$s_{ik} =\frac{1}{n} \sum _{j=1}^{n}(x_{ji} -\bar{x}_{i} )(x_{jk} -\bar{x}_{k}  )$;  i=1,2,..,p and k=1,2,\… ,p.

It measures the association between the kth variable.

Variance is the most commonly used measure of dispersion (variation) in the data and it is directly proportional to the amount of variation or information available in the data.

Sample Correlation Coefficient

The sample correlation coefficient for the ith and kth variable is

\[r_{ik} =\frac{s_{ik} }{\sqrt{s_{ii} } \sqrt{s_{kk} } } =\frac{\sum _{j=1}^{n}(x_{ji} -\bar{x}_{j} )(x_{jk} -\bar{x}_{k} ) }{\sqrt{\sum _{j=1}^{n}(x_{ji} -\bar{x}_{i} )^{2}  } \sqrt{\sum _{j=1}^{n}(x_{jk} -\bar{x}_{k}  )^{2} } } \]
$\mbox{ where } i=1,2,..,p \mbox{ and}  k=1,2,\dots ,p$

Note that $r_{ik} =r_{ki} $ for all $i$ and $k$, and $r$ lies between -1 and +1. $r$ measures the strength of the linear association. If $r=0$ the lack of linear association between the components exists. The sign of $r$ indicates the direction of the association.