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Standard Deviation – Formula, Examples, Symbol, Calculations

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19th Feb, 2024
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    Standard Deviation – Formula, Examples, Symbol, Calculations

    Standard deviation, also known as SD or indicated by the symbol "σ," indicates how far a value deviates from the mean value. A low standard deviation indicates that the values are often within a few standard deviations of the mean. In contrast, a large standard deviation indicates that the values exceed the mean. Before calculating the standard deviation, there are many things to know.  

    Calculating standard deviation in excel is quite a task since there are many things to remember. For example, there are different types of data grouped or ungrouped, and the formula for standard deviation also changes. Since learning so many different things from different sources will increase the complexity so, to ease the situation, here is a guide that covers the most important topics related to the standard deviation. Take PMP training course and excel in this field. 

    What is Standard Deviation?

    Are you looking for a standard deviation definitionIn descriptive statistics, the standard deviation measures how widely distributed or scattered the datasets are about their mean. The measure of the variation of the data sets from the mean reveals how the data are distributed across the given data. The square root of a sample's variance represents the standard deviation of a statistical population, random variable, data collection, or probability distribution. 

    The average deviation of the computing mean is calculated as x1, x2...xn if there are n observations with the values x1, x2...xn. Nevertheless, the sum of the squares of departures from the mean will not appear to be an accurate indicator of dispersal. The observations xi are likely close to the mean x if the mean of the squared differences from the mean is low.  

    This dispersion is at a lower level. If this total is high, there is a greater deviation between the observations and the mean x. As a result, we conclude that ∑(xi−x̄)2 is a reliable indication of the level of scatter or dispersion. 

    Standard deviation is among the most essential and crucial risk measures used by researchers, portfolio managers, or consultants. The standard deviation of mutual funds, as well as other products, is disclosed by investment groups. A significant dispersion reveals how far the investment return deviates from the projected average return. This data is frequently given to investors and end users since it is simple to grasp. 

    Why is Standard Deviation Important?

    There are many reasons why the standard deviation is considered important. Some of them are given below: 

    1. Whenever the data is dispersed, it makes the results easier to grasp. 
    2. A standard deviation of distribution or the dataset will indeed be higher the more evenly dispersed the dataset is. 
    3. Corporate leaders utilize the standard deviation in excel for finances to comprehend threat management and to make wiser investment choices. 
    4. It facilitates the computation of the error margins typically appearing in survey results. 

    Applications of Standard Deviation

    Standard deviation in statistics is used in many operations, which is one of the reasons many universities and school boards prefer this topic in the syllabus rather than any other thing. Here are some applications of standard deviations. 

    1. Marketers frequently compute the standard deviation of revenue generated per advertisement to determine how much standard deviation and variance in earnings to anticipate from a specific ad. 
    2. Managers of human resources frequently determine the standard deviation of earnings in a particular industry, so they may decide what kind of salary variation to provide to recruits. 
    3. To comprehend how much variety there is in the ages of the people, they offer insurance; insurance researchers frequently determine the standard deviation of that age. 
    4. Teachers, trainers, or professors can determine which classes had the most variety in exam results among students by calculating the standard deviation of exam results for different classes. 

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    SD Formula 

    Standard deviation is one of the vastest topics of engineering alone. Many students faced a lot of difficulty with all the aspects of standard deviation alone. And also, the standard deviation is later used in many things like measures of dispersion, making it even more complex. But truth be told, calculating the standard deviation is not that hard, even if it is the easiest thing a person ever calculates. No doubt some problems took hours of calculations but apart from that, everything works just fine. 

    The easiest way to calculate standard deviation is simply to calculate the mean of the data provided. Now just compute the variance, which is the second last step in calculating the standard deviation. After that, all values are placed in the formula given below. So let us understand how to find Standard Deviation formula. 

    Formula for Calculating Standard Deviation

    In simple theory, the standard deviation can easily be calculated by the square root of variance. To calculate the variance, a person needs to calculate the mean first. Keeping this in mind, here is a sample standard deviation formula to calculate the standard variance. The standard deviation symbol is σ.  

    • x̄ = The dataset's mean result 
    • xi = the ith point in the dataset's value 
    • n= Data set size in terms of data points 

    How is Standard Deviation Calculated?

    Steps to Calculate Standard Deviation

    Since the calculation of the mean is compulsory and used much with sample standard deviation, many experts prefer to calculate the mean first while calculating the standard deviation. But this decision will change as people have different preferences, either to calculate the mean separately or while calculating the standard deviation. So we just leave that decision to the reader. Check below how to find standard deviation: 

    1. Find the average of all the data points. Then, the sum of all data points is divided by the total number of data points to determine the mean.   
    2. Determine each data point's variance. Then, the variance is obtained for each of them by deducting the mean from the value for every data point.   
    3. Square every data point's variance (Step 2)  
    4. Sum of the variance values by squares (Step 3).   
    5. The number of observations in the data set less than 1 is equivalent to the sum of the squared variance values (Step 4). 
    6. Determine the quotient's square root (Step 5).   

    Standard Deviation of Grouped Data

    When there is grouped data or a grouped frequency distribution, the frequency of the data values can be used to calculate the standard deviation. Let us take an example to understand the standard deviation formula for grouped data better: 

    Example:  For the following data, determine the mean, variance, and standard deviation: 

    Class Interval0-1010-2020-3030-4040-5050-60
    Frequency27107542

    Solution :

    Class IntervalFrequency (f)Mid Value (xi)fxifxi2
    0-10275135675
    10-2010151502250
    20-307251754375
    30-405351756125
    40-504451808100
    50-602551106050

    ∑f=55
    ∑fxi= 925∑fxi2= 27575
    • N = ∑f = 55 
    • Mean = (∑fxi)/N = 925/55 = 16.818 
    • Variance = 1/(N – 1)[∑fxi2 – 1/N(∑fxi)2] 
    • = 1/(55 – 1) [27575 – (1/55) (925)2] 
    • = (1/54) [27575 – 15556.8182] 
    • = 222.559 
    • Standard deviation = √variance = √222.559 = 14.918 

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    Standard Deviation of Ungrouped Data 

    For various sets of data, different standard deviation methods are used. The dispersion measures data's divergence from the mean or average location. The standard deviation can be discovered using one of two methods: 

    1. Actual Mean Method

    The formula for the actual mean method will be:

    • σ = √(∑x−x̄)2 /n) 
    • Now, let's take an example for a better understanding of 3, 2, 5, and 6.  
    • As stated above, using the formula for calculating the mean, it'll be calculated as 16/4 =4. 
    • The squared differences from average mean = (4-3)2+(2-4)2 +(5-4)2 +(6-4)2= 10 
    • Similarly, Variance = Squared differences from mean/ number of data points =10/4 =2.5 
    • Now, using the formula above, Standard deviation = √2.5 = 1.58 

    2. Assumed Mean Method

    When the x values are big, the mean is determined by selecting an arbitrary value (A). It is also sometimes called the Assumed mean instead of the arbitrary value. The deviation from this assumed mean is calculated as d = x - A. 

    • σ = √[(∑(d)2 /n) - (∑d/n)2] 

    Standard Deviation of Random Variables

    A spread metric for a random variable distribution that identifies how far the values deviate from the expected value. 

    It is common to write or X as the random variable X's standard deviation. 

    When dealing with discrete random variables, the standard deviation is determined by adding the square of the difference between the random variable's value and the expected value, the affiliated probability of the irregular variable's value, multiplied by each of the random variable's values, and taking the square root of the sum. 

    • The equation: σ  
    • The variance, Var(X) = σ 2, is equal to the square of the standard deviation. 

    Standard Deviation Example

    Now, let us figure out the standard deviation for the number of gold coins on a ship. 

    On the ship, there are a total of 100 people. According to statistics, 100 people make up the population. Therefore, if you know each person's total number of gold coins, you can utilize the standard deviation equation for the overall population. 

    Let us take a 5-person sample as an example and apply the standard deviation equation to this group. With a sample size of 5, you can use the standard deviation equation to calculate a population sample. 

    Assume that five people have the number of gold coins: 4, 2, 5, 8, and 6. 

    • Mean : 
    • x̄ = sum of observation / total number of observations 
    • = 4+2+5+8+6 / 5 
    • = 5 

    Now xn-x̄ for each value given in the dataset 

    • 4-5 = -1 
    • 2-5 = -3 
    • 5-5 = 0 
    • 8-5 = 3 
    • 6-5 = 1 
    • ∑(x−x̄)2=  (-1)2+ (-3)2+02+32+12 
    • = 20 

    Standard Deviation 

    • σ = √[ ∑(x - x̄)2 / N ] 
    • = √20/4 
    • = √5 
    • = 2.236 
    • = 2.236 

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    Conclusion

    The above article covers most of the topics directly concerned with the standard deviation, average standard deviation, and much more. Since there are many basic things such as continuous or discrete data etc. The article does not cover some obvious terms, such as mean-variance and others. The standard deviation is not a small thing. It is just a small part of a very big concept.  

    Engineering and computer science students often get this chapter in their curriculum. Students, Feel free to ask any questions in the comments. The data analyst job is one of the common professions that a person can do by mastering the concept of standard deviation and other related topics. You can always go for KnowledgeHut PMP training course to better understand statistics and its implementation in real life. 

    Frequently Asked Questions (FAQs)

    1. What does a standard deviation of 1 mean?

    A standard deviation is a statistical measure of variance in a population or group. A standard deviation of one indicates that 68% of the population is within plus or minus the standard deviation from the average. For example, assume the average male height is 5 feet 9 inches, and the standard variation is three inches. Then 68% of all males are between 5' 6" and 6', 5'9" plus or minus 3 inches.  

    2. What is a good standard deviation?

    While considering the graphical representation, the data set near the mean is considered good. Considering just the calculation of standard deviation, the coefficient of variance or CV whose value relies on CV < 1 is considered a good standard deviation.  

    3. How do you tell if a standard deviation is high or low?

    One of the easiest ways to determine whether the standard deviation is high or low is through graphical representation. As stated above, calculating standard deviation is not as difficult as people say. And depicting the standard deviation through the diagram is even easier and helps a person determine if the standard deviation is high or low. In general, if the coefficient of variance or CV >= 1, it is high; otherwise, it is low. The coefficient of variance is calculated as the standard deviation divided by the mean. 

    4. Which is better, high or low standard deviation?

    As given the definition above, the standard deviation shows how further the data move from the mean. There are two things mainly happening in this which is either the high, which means data shows the data is spread very far from the mean and also is considered as not reliable another is low, which shows the data is close to the mean, which is considered as the best or more reliable.

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