About Standard Deviations
The Standard Deviation is the positive square root of the variance. One of the most basic approaches of Statistical analysis is the Standard Deviation. The Standard Deviation, abbreviated as SD and represented by the letter ", indicates how far a value has varied from the mean value. A low Standard Deviation indicates that the values are close to the mean, whereas a large Standard Deviation indicates that the values are significantly different from the mean. Let's look at how to determine the Standard Deviation of grouped and ungrouped data, as well as the random variable's Standard Deviation.
What is Standard Deviation?
In descriptive Statistics, the Standard Deviation is the degree of dispersion or scatter of data points relative to the mean. It is a measure of the data points' Deviation from the mean and describes how the values are distributed over the data sample. The Standard Deviation of a sample, Statistical population, random variable, data collection, or probability distribution is the square root of the variance.
When we have a certain amount of observations and they are all different, the value's mean Deviation from the mean is then calculated.
On the other hand, the sum of squares of deviations from the mean does not appear to be a reliable measure of dispersion. When the average of the squared differences from the mean is low, the observations are close to the mean. This is a less dispersed level of dispersion. If this number is large, it implies that the observations are dispersed from the mean to a greater extent.
Standard deviation is the measurement of the dispersion of the data set from its mean value. It is always measured in arithmetic value. Standard deviation is always positive and is denoted by σ (sigma). Standard Deviation is very accurate and is preferred from other measures of dispersion.
The Standard Deviation is calculated as The square root of variance by determining each data point's deviation relative to the arithmetic mean. In case the data-points are far from the mean, it denotes a higher deviation within the set of data. Hence, it indicates more spread out the data, the higher is the standard deviation. The formula to calculate Standard Deviation is:
s = \[\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n-1}}\]
where:
x(i) = value of the i’th point in the set of data
x(bar) = the mean-value of the set of data
n = the number of data-points in the set of data
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Properties of Standard Deviation
Standard Deviation is only used in measuring dispersion or spread around the mean value of the data set.
Standard deviation is always in positive value.
It determines the dispersion or variation that exists from the average value.
Standard deviation is a very sensitive outlier. Any single outlier can distort the picture of dispersion.
For the data set with an approximately same mean value, the greater the dispersion or spread, the greater the Standard deviation.
Standard deviation is zero when the values of a particular data set are the same.
While analyzing the normally distributed data, the Standard Deviation is used in conjunction along with the mean to calculate the data intervals.
If \[\overline{x}\] = mean, S = Standard Deviation, and x = Value in the Data set, then
around 68% of the Data is in the interval:- \[\overline{x}\] - S < x < mean + S.
around 95% of the Data is in the interval:- \[\overline{x}\] - 2S < x <mean + 2S.
around 99% of the Data is in the interval:- \[\overline{x}\] - 3S < x < mean + 3S.
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Standard Deviation Calculation
Before calculating the Standard Deviation, it is essential to underline the three types of data distribution. These are:
Individual Series
A single column denoting the observation is available here.
Discrete Series
Two columns represent different data. One column shows the observation, while the other column is for frequency corresponding to the observation column.
Frequency Distribution
It has two columns, one representing the observations, and the other is corresponding frequency.
Here the observations are classified further into intervals or classes.
Sigma for Individual Series
Three methods can calculate the Standard deviation for individual series; these are:
A Direct Method to Calculate Standard Deviation
Use the formula ∑X/N to calculate the arithmetic mean. After this, we calculate the deviations of all the observations from the mean value using the formula D= X-mean.
Now, the deviations, x, are squared and summed. The resultant value is then divided by the total number of observations. The square root of the above-derived value = Standard deviation
The formula is - σ = √[∑D²/N]
Here, D = deviation of an item that is relative to mean. It is calculated as D = X- mean.
N = Number of observations
Short-Cut Method
In this method, any random value is assumed to calculate deviation. It is believed that the assumed value is in the Middle of the Range of Values. The short cut method is derived using the formula;
σ = √[(∑D²/N) – (∑D/N)²]
Step-Deviation Method
It is a simple form of the short-cut method. Here, we select a common factor C, among the deviations. All the deviation values reduce when divided by C, simplifying the calculations. The formula is;
Standard deviation D (σ)= √[(∑D’²/N) – (∑D’/N)²] × C
D'= step-deviation of Observations relative to an Assumed mean. It is calculated as D'= (X-A)/C
C= Common Factor chosen.
Sigma for Discrete Series
There are two ways to calculate Standard Deviation in discrete series, theses are:
Direct Method
We know that in the discrete series, another frequency column is added; the direct method formula to calculate SD is:
Standard deviation (σ) = √(∑fD²)/N)
Short-Cut Method
Standard deviation (σ) = √[(∑fD²/N) – (∑fD/N)²]
Sigma for Frequency Distribution
Three different methods can be used to calculate standard deviation in frequency distribution series; these methods are:
Direct Method
The direct method employed to derive standard deviation in a frequency distribution is very similar to the discrete series done above. The value of observation (when used) in the frequency distribution is the only difference between the two series. Here, the mid-value of the class is determined by dividing the sum' of the upper value of the class and the lower value of the class. The value thus derived is used for calculation. The formula is;
Standard Deviation (σ) = √(∑fD²)/N)
In the calculation, D = Deviation of an item that is relative to mean value and is calculated as,
D = Xi – Mean
F = frequencies corresponding to the Observations
N = The summation of the frequency.
Step-Deviation Method
The step-deviation method is the shortcut method to determine the Standard Deviation. The formula is
Standard Deviation (σ) = √[(∑fD’²/N) – (∑fD’/N)²] × C
In the above calculation, D'= Step-Deviation of the observations relative to the assumed value. It is calculated as- D'= (Xi-A)/C
N = The Summation of Frequency.
C = Common Factor chosen
Did You Know?
Without Standard Deviation D, one can't compare two sets of data effectively. Suppose there are two data sets having the same average. Does that imply that the sets of data are exactly the same? No. For ex. the data sets - 199, 200, 201, and other 0, 200, 400 have the same 200 average, yet they have different standard deviations. Here, the first data has a small standard deviation (s=1) in comparison to the second set of data (s=200).
FAQs on Standard Deviation: Calculation and Applications
1. What is Standard Deviation and why is it considered a crucial measure in statistics?
Standard Deviation (often denoted by the Greek letter sigma, σ) is a statistical measure that quantifies the amount of dispersion or spread of a set of data values from their mean (average). It is crucial because it provides a standardised way to understand data consistency. A low standard deviation indicates that the data points are clustered closely around the mean, suggesting high consistency, while a high standard deviation indicates that the data points are spread out over a wider range, suggesting low consistency.
2. How is the Standard Deviation of a data set calculated?
The calculation of Standard Deviation involves a few key steps. As per the CBSE/NCERT curriculum, the general process is as follows:
Step 1: Calculate the mean (average) of all the data points in the set.
Step 2: For each data point, calculate its deviation by subtracting the mean from the data point's value.
Step 3: Square each of these deviations to make them positive.
Step 4: Calculate the average of these squared deviations. This value is known as the Variance.
Step 5: Find the square root of the variance. The result is the Standard Deviation.
3. What is the fundamental difference between Standard Deviation and Variance?
The fundamental difference lies in their units and interpretation. While both measure data dispersion, Standard Deviation is the square root of the Variance. The key distinction is:
Units: Standard Deviation is expressed in the same units as the original data (e.g., rupees, kilograms, or marks), making it more intuitive to interpret. Variance is expressed in squared units (e.g., rupees squared), which is harder to relate directly to the dataset.
Usage: Standard Deviation is preferred for describing the spread of data, while variance is crucial for more advanced statistical calculations and analyses like ANOVA.
4. How should one interpret a high or low Standard Deviation in a practical scenario?
Interpreting Standard Deviation helps in making decisions. For example, in business:
A low Standard Deviation in a product's sales figures per month suggests stable and predictable sales. This is often desirable for inventory and cash flow management.
A high Standard Deviation in the same sales figures indicates volatile and unpredictable sales. This signifies higher risk and might require more flexible strategies for marketing and stock management.
Essentially, low SD means consistency and reliability, while high SD means variability and risk.
5. Can you provide a real-world example of how Standard Deviation is used in commerce?
In finance, a key application of Standard Deviation is to measure the volatility or risk of an investment. An investment manager comparing two stocks might find that:
Stock A has an average annual return of 15% with a Standard Deviation of 5%.
Stock B has an average annual return of 15% with a Standard Deviation of 20%.
Even though both stocks have the same average return, Stock B is considered much riskier because its returns are more spread out (more volatile). A risk-averse investor would likely prefer Stock A due to its lower Standard Deviation and more predictable performance.
6. What are the main advantages and disadvantages of using Standard Deviation?
Standard Deviation is a powerful tool, but it has its own merits and demerits.
Advantages:
It is rigidly defined and based on all observations in the dataset.
It is the most reliable measure of dispersion and is highly suitable for further algebraic treatment.
It is less affected by sampling fluctuations compared to other measures.
Its calculation can be complex and time-consuming without a calculator.
It gives more weight to extreme values (outliers) because deviations are squared, which can sometimes misrepresent the overall dispersion.
7. Why can the Standard Deviation value never be negative?
Standard Deviation can never be negative due to the mathematical process used to calculate it. The core of the calculation involves squaring the deviations from the mean. Squaring any number, whether it is positive or negative, always results in a positive value. The sum of these positive squared deviations will be positive, and its average (the variance) will also be positive. The final step is taking the square root of the variance, which by convention is always the positive root. Therefore, the Standard Deviation is always a non-negative number.
8. What are the different methods for calculating Standard Deviation for grouped data as per the NCERT syllabus?
For grouped data (discrete or continuous series), the NCERT syllabus outlines three main methods for calculating Standard Deviation:
Direct Method: This method uses the actual mean to calculate deviations. It is straightforward but can be tedious if the mean is a decimal.
Short-cut Method (Assumed Mean Method): This simplifies calculations by using an 'assumed mean'. Deviations are calculated from this assumed value, and a correction factor is applied in the final formula.
Step-Deviation Method: This further simplifies the short-cut method by dividing the deviations by a common factor 'c'. This is especially useful when class intervals are of equal size, as it makes the calculations much smaller and easier to manage.
9. In what situation might another measure of dispersion, like Mean Deviation, be preferred over Standard Deviation?
While Standard Deviation is mathematically superior, Mean Deviation might be preferred in specific contexts. One key situation is when a simpler, more intuitive measure is needed that is less sensitive to extreme outliers. Because Standard Deviation squares the deviations, it gives disproportionately large weight to outliers. Mean Deviation uses the absolute values of deviations, which prevents this magnification. Therefore, if a dataset contains significant outliers that are not representative of the general data, Mean Deviation can sometimes provide a more realistic measure of the typical spread.
