So, to overcome this we can make use of a separate static method within the same class. Standard deviation is an important topic of statistics. Mention Some Basic Points on Difference Between Standard Deviation and Variance? After gathering these values the same steps as mentioned above are to be followed. The standard error of the mean can be determined as the standard deviation of such a sample means including all the possible samples drawn from the same population. When the data points are grouped, we first construct a frequency distribution. Now, we have to find the standard deviation of the first 5 natural numbers. It gives an estimation of how individuals in data are dispersed from the mean value. 2. However, their standard deviations (SD) differ from each other. The formula for the relative standard deviation is given as: RSD = \[ \frac{s \times 100} {\text{X bar}}\]. Sample B is more variable than Sample A. This means it gives you a better idea of your datas variability than simpler measures, such as the mean absolute deviation (MAD). Determine the standard deviation of the first 5 natural numbers. A high standard deviation means that values are generally far from the mean, while a low standard deviation indicates that values are clustered close to the mean. The number of successes is a random variable in a binomial experiment. Step 1: Determine the mean of the observations, i.e. The spread of statistical data is measured by the standard deviation. Your Mobile number and Email id will not be published. How do you calculate the standard deviation? According to Spiegel, The Standard Deviation is the square root of the arithmetic mean of the squares of all deviations. The square root of the variance is the Standard Deviation of a random variable, sample, population, data collection, or probability distribution. 1. The squared differences from mean = (4-3)2+(2-4)2 +(5-4)2 +(6-4)2= 10, Variance = Squared differences from mean/ number of data points =10/4 =2.5. A z-score tells you how many standard deviations away an individual data value falls from the mean. When we have n number of observations and the observations are \(x_1, x_2, ..x_n\), then the mean deviation of the value from the mean is determined as \(\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\). Different formulas are used for calculating standard deviations depending on whether you have collected data from a whole population or a sample. This below solved example problem for frequency distribution standard deviation may help the users to understand how the values are being used to workout such calculation based on the above mathematical formulas.Example Problem:In a class of students, 9 students scored 50 to 60, 7 students scored 61 to 70, 9 students scored 71 to 85, 12 students scored 86 to 95 and 8 students scored 96 to 100 in the subject of mathematics. The experimental probability consists of many trials. Grouped data standard deviation calculator - step by step calculation to measure the dispersion for the frequency distribution from the expected value or mean based on the group or range & frequency of data, provided with formula & solved example problems. November 11, 2022. Answer:The standard deviation of the probability distribution is 1.45. Calculate the squared deviations from the mean. Standard deviation is the positive square root of the variance. CV = s / x. where: s: The standard deviation of dataset. According to the definition of standard deviation, when the standard deviation of a series is 0, it means that all of the values in the series are equal to the mean, making all deviations zero, and hence, the standard deviation is also zero. sum+=Math.pow((input[i]-mean),2); mean=sum/(n-1); Any value predetermined decreases the efficiency of code. 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Variance is nothing but average taken out from the standard deviation. The population standard deviation formula looks like this: When you collect data from a sample, the sample standard deviation is used to make estimates or inferences about the population standard deviation. To find the standard deviation, we take the square root of the variance. It is the most popular method of determining standard deviation.The steps to calculate the standard deviation of a frequency distribution series by the Step-Deviation Method are as follows: Example: Calculate the standard deviation of the following series by Step-Deviation Method. Morningstar Office uses 14 asset classes in the Efficient Frontier methodology and 12 asset classes in the goal planner (all those listed below except Commodities and Real Estate.) So our result Standard Deviation is of double type as well. ), Calculate the square root of the variance. Step 2: Calculate the squared deviations from the mean, i.e. It tells you, on average, how far each value lies from the mean. The population standard deviation formula is given as: Similarly, the sample standard deviation formula is: \(\bar x\) = Arithmetic mean of the observations. by First of all, a value is assumed from the mid-values of the given data set, and then the deviations of the assumed value are taken from the mid-values. Standard Deviation is the measure of the dispersion of data from its mean. (Data value Mean)2. Thus we conclude that \(\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\) is a reasonable indicator of the degree of dispersion or scatter. Calculate Xs variance and standard deviation. Using Scanner Class in Java, inputs can be read at runtime and can be given according to the user/tester and can always be changed without the strain of going through the code. Larger the deviation, further the numbers are dispersed away from the mean. The standard deviation is the measure of dispersion or the spread of the data about the mean value. Standard Deviation is considered to be the best way of determining the dispersion of a data set. The frequency distribution standard deviation formula along with the solved example let the users to understand how the values are being used in this calculation. Step 2: Calculate the squared deviations from the mean, i.e. A few plants were selected randomly and their heights in cm were recorded as follows: 51, 38, 79, 46, 57. To learn more Maths-related concepts quickly, download BYJUS The Learning App and explore more videos. If this number is large, it implies that the observations are dispersed from the mean to a greater extent. The standard deviation, on the other hand, is the range of data values around the mean. Step 1: Compute the mean for the given data set. Class Deviations. The deviation is denoted by d (d = m A). In a class of 50 pupils, four students were chosen at random, and their final evaluation scores were recorded as 812, 982, 836 and 769. The standard deviation is effectively the square root of the variance. The sample standard deviation formula is: \(s=\sqrt{\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}\), where \(\bar x\) is the sample mean and \(x_i\) gives the data observations and n denotes the sample size. In this method, the standard deviation of a series of data is determined directly through deviations of the arithmetic mean from each value. Standard deviation formulas for populations and samples, Steps for calculating the standard deviation. SEM is basically an approximation of standard deviation, which has been evaluated from the sample. Prove that the first n natural numbers standard deviation equals \(\begin{array}{l}\sigma = \sqrt{\frac{n^{2}-1}{12}}.\end{array} \). In the above variance and standard deviation formula: With the help of the variance and standard deviation formula given above, we can observe that variance is equal to the square of the standard deviation. Standard deviation is one of the basic methods of statistical analysis. Variance is simply stated as the numerical value, which mentions how variable in the observation are. The measures of central tendency (mean, mode, and median) are exactly the same in a normal distribution. We can easily calculate variance as the square of standard deviation if we know how to calculate standard deviations. Revised on For data with almost the similar mean, the larger the spread, the greater the value of standard deviation. A high standard deviation means the data points are more spread out from the average, while a low standard deviation means the points are closer to the mean. Both measures reflect variability in a distribution, but their units differ: Although the units of variance are harder to intuitively understand, variance is important in statistical tests. Example 3: Find the standard deviation of X which has the probability distribution as shown in the table below. (The data value - mean), Find the average of the squared differences. Its useful for complex statistical calculations like comparing two data sets variability. The procedure to use the standard deviation calculator is as follows: Step 1: Enter the numbers separated by a comma in the respective input field. As a result, the required probability distribution looks like the following: Now, substitute the values on the formula. December 10, 2022 A small Standard Deviation means the results are close to the mean, whereas a big Standard Deviation means the data are widely divergent from the mean. The standard deviation formula we apply depends on whether the data is regarded as a population on its own or a sample representing a larger population. Why is standard deviation a useful measure of variability? In Mathematical terms, standard dev formula is given as: is the sample variance, m is the midpoint of a class. This is a less dispersed level of dispersion. There are six main steps for finding the standard deviation by hand. The standard deviation of a random variable with a binomial distribution is: = npq, where mean: = np, n = number of trials, p = probability of success, and 1-p =q represents the probability of failure. The compiler has also been added so that you can execute the programs yourself. During a survey, 6 students were asked the number hours per day they give time to their studies on an average? Standard Deviation formula to calculate the value of standard deviation is given below: Standard Deviation Formulas For Both Sample and Population, \[\sigma = \sqrt{\frac{\sum (X - \mu)^{2}}{n}} \], \[s = \sqrt{\frac{(X - \overline{X})^{2}}{n - 1}} \], Notations For the Sample Standard Deviation Formula and Population Standard Deviation Formula. Lets take two samples with the same central tendency but different amounts of variability. The predicted value of the experiment, denoted by, is known as this mean. The formula for mean is: Mean=Sum/Total Number. (Mean of the data value)2, Calculate the mean of the squared differences. If X is a random variable, the standard deviation is determined by taking the square root of the sum of the product of the squared difference between the random variable, x, and the expected value () and the probability associated value of the random variable. As a result, the sum of the square of the first n natural number is: Thus, first n natural numbers standard deviation. The standard deviation is usually calculated automatically by whichever software you use for your statistical analysis. In such a scenario, if something as important as input is fixed then the code is not at its optimal state. The deviation squares (d 2) are then-multiplied by the frequency (f) of the respective classes to get (fd 2). It can be calculated in three different series; viz., Individual, Discrete and Frequency Distribution Series. Therefore, a population of the sampled means will appear to have different variance and mean values. 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Both measures reflect variability in distribution, but their units differ: Standard deviation is expressed in the same units as the original values (e.g., minutes or meters). Solution: In simple terms, the CV is the ratio between the standard deviation and the mean. While storing these values, they are converted into integers or long type based on requirement. Then the same standard deviation formula is applied. Published on 3. 2. i don't know what the statistical details of std. Variance - The variance is a numerical value that represents how broadly individuals in a group may change. We get. The last step is to use the formula of the direct method to calculate the standard deviation. Example: Calculate the standard deviation of the following series by Direct Method. 2. Short-Cut Method In this method, the standard deviation of a series of data is determined by obtaining deviations of the mid-values of the class intervals of a data set. The difference between standard deviation and variance is given below in tabulated form: 8. It is scale-independent but not origin-independent. Suppose that the entire population of interest is eight students in a particular class. Which of the Following Is the Measure of Variability? Java Program to calculate Index Multiplier. While this is not an unbiased estimate, it is a less biased estimate of standard deviation: it is better to overestimate rather than underestimate variability in samples. However, Reference Links Are Allowed To Our Original Articles - JT. The standard deviation is the average amount of variability in your dataset. After this the mean has to be found. The squared differences from mean = (4-3)2+(2-4)2 +(5-4)2 +(6-4)2= 10, Variance = Squared differences from mean/Total number of data. Hence, the standard deviation (S.D) of their heights is 15.5. It actually measures the amount of variation of a specific set of values. The standard deviation should only be used when the population being studied has an equal number of data points on either side of the mean. This is a lower degree of dispersion. from https://www.scribbr.com/statistics/standard-deviation/, How to Calculate Standard Deviation (Guide) | Formulas & Examples. Standard deviation is speedily affected outliers. For example: 418.6 951.9 820.3 951.0 651.8 635.8 538.7 842.9 -194.0 487.1 484.5 840.5 661.4 Calculate How to enter data as a frequency table? Here the mean of these data points is 16/4 = 4. When you have collected data from every member of the population that youre interested in, you can get an exact value for population standard deviation. To find the expected value of X, find the product X. P(X) and sum these terms. The higher is the dispersion or variability of data, the larger will be the standard deviation and the larger will be the magnitude of the deviation of value from the mean whereas the lower is the dispersion or variability of data, the lower will be the standard deviation and the lower will be the magnitude of the deviation of value from the mean. How to Calculate Standard Deviation (Guide) | Formulas & Examples. It is important to notice similarities between the variance of sample and variance population. By using our site, you The list of standard deviation v/s variance is given below in tabulated from. Multiply each deviation from the mean by itself. The standard deviation = 2.06 = 1.43, Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. Standard deviation is simply stated as the observations that are measured through a given data set. In Mathematical terms, standard dev formula is given as: The standard error of the mean is a procedure used to assess the standard deviation of a sampling distribution. The following plants were chosen at random, and their heights were recorded in cm: 38, 51, 46, 79, and 57. dev. Sample standard deviation formula = \(\sigma=\sqrt{\frac{1}{N-1} \sum_{i=1}^{N}\left(X_{i}-\mu\right)^{2}}\) and variance formula = 2 = (xi x)2/(n-1), Variance is the sum of squares of differences between all numbers and meanswhere is Mean, N is the total number of elements or frequency of distribution. Step 3: Calculate the squared differences average, i.e. The standard deviation of a dataset is a measure of its dispersion related to its mean. It is also called a coefficient of variation. Variance is expressed in much larger units (e.g., meters squared). The weight of each egg laid by hen is given below. For example, if you have 5 people with 3 Standard deviation = Variance. Feedback on this topic? but generally it's a good rule of thumb in figuring out your performance on a curved exam. For n observations in the sample, find the mean of them. By squaring the differences from the mean, standard deviation reflects uneven dispersion more accurately. Suitable examples and sample programs have been included in order to make you understand simply. It is denoted by a Greek Symbol (sigma). Pritha Bhandari. Standard Deviation = 10.89 2. When we have a certain amount of observations and they are all different, \[x_{1},x_{2},x_{3},x_{4},x_{5}x_{n}\], then the value's mean deviation from the mean is calculated as, \[\sum_{i=1}^{n} (x_{i} - \overline{x})^{2}\]. Standard Deviation questions and answers can help students learn the concepts fast. Sample mean is represented by the symbol. It is also known as root mean square deviation.The symbol used to represent standard deviation is Greek Letter sigma ( 2). Here are two standard deviation formulas that are used to find the standard deviation of sample data and the standard deviation of the given population. Subtract the mean from each score to get the deviations from the mean. In all the above methods, the entire code was written in the main method only. It is dependent on all values, and so contains data for the entire series. In order to arrive at the standard deviation of the set of numbers we have step wise process. This is denoted by X, Y, or Z, as it is a function. Calculate the standard deviation. If the frequency distribution is continuous, each class is replaced by its midpoint. (Standard deviation = Variance), \(\sigma=\sqrt{\frac{1}{N} \sum_{i=1}^{N}\left(X_{i}-\mu\right)^{2}}\), \(s=\sqrt{\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}\), In a binomial experiment, the number of successes is a random variable. The answers of the students are as follows: 2, 6, 5, 3, 2, 3. Standard Deviation - On the other hand, standard deviation perceives the significant amount of dispersion of observations when comes up close with data. Under this method, the deviation of values is taken from the arithmetic mean of the given set of data. Step 4: To find the standard deviation, calculate the variances square root, i.e. Have questions on basic mathematical concepts? It is the quantity which expresses the variation of the group from the mean value. In this method, the standard deviation of a series of data is determined by obtaining deviations of the mid-values of the class intervals of a data set. so if the class average is a 75, and standard deviation is 10, and the class average is by definition a C+/B-, then an 85 is like one grade higher, a B+/A-, and so on. Sum them up and find the square root of the average of the squared differences. Learn the why behind math with our certified experts, Standard Deviation of Probability Distribution, Find the squared differences from the mean. If all values in a given set are similar, the value of standard deviation becomes zero (because each value is equivalent to the mean). Determine their standard deviation. (Data value Mean) 2. Scribbr. 200. The measure of the dispersion of data points relative to the mean is defined by the standard deviation in descriptive statistics. 2. Problem 3: Find the standard deviation of X if the pdf of X is given by the function p(x) = (1 x) for x [0, 1]. Most values cluster around a central region, with values tapering off as they go further away from the center. For a finite set of numbers, the population standard deviation is found by taking the square root of the average of the squared deviations of the values subtracted from their average value. However, for that reason, it gives you a less precise measure of variability. Mean = Sum of all observations / Number of observations. Using the actual mean method, calculate the standard deviation for the data 3, 2, 5, and 6. The sample mean is the average and is calculated as the addition of all the observed outcomes from the sample divided by the total number of events. Sample Mean (X) = (812+836+982+769)/4 = 849.75, Variance = \( \dfrac{\sum^{N}_{i=1} (X_i - \bar{X})^2}{N-1} \), =\( \dfrac{\sum^{4}_{i=1} (X_i - 849.75)^2}{3} \), = [(812 - 849.75)2 + (836 - 849.75)2 + (982 - 849.75)2 + (769 - 849.75)2] /3 = 92.4, Standard Deviation = 92.4 = 2 23.1 = 9.6, Answer: Standard Deviation for this data is 9.6. A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. The formula for standard deviation becomes, \[ \sqrt{\frac{1}{N} \sum\limits_{i = 1}^{n} f_{i}(x_{i} - \overline{x})^2 }\]. Therefore, the standard deviation is 1.58. When you have the standard deviations of different samples, you can compare their distributions using statistical tests to make inferences about the larger populations they came from. Step 3: Finally, the mean, variance, and standard deviation for the given set of data will be displayed in the output field. Variance, \[\sigma^{2} = \frac{\sum_{i=1}^{n} (x_{i} - \overline{x})^{2}}{n} \], Standard Deviation, \[\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_{i} - \overline{x})^{2}}{n}} \]. Standard deviation is the concept that takes into account the random and subjective nature of the perspectives of people What are the Different Properties of Standard Deviation? Step by step calculation: Follow these below steps using the above formulas to understand how to calculate standard deviation for the frequency table data set step 1: find the mid-point for From learning that SD = 13.31, we can say that each score deviates from the mean by 13.31 points on average. For n number of observtions, \(x_1, x_2, ..x_n\), and the frequency, \(f_1, f_2, f_3, f_n\) the standard deviation is: \(\sigma=\sqrt{\frac{1}{N} \sum_{i=1}^{N}f_i \left(X_{i}-\bar x\right)^{2}}\). Although there are simpler ways to calculate variability, the standard deviation formula weighs unevenly spread out samples more than evenly spread samples. If we get a low standard deviation then it means that the values tend to be close to the mean whereas a high standard deviation tells us that the values are far from the mean value. Standard deviation is the positive square root of variance. The curve with the lowest standard deviation has a high peak and a small spread, while the curve with the highest standard deviation is more flat and widespread. Hence, the standard deviation is calculated as, Population Standard Deviation - \[\sigma = \sqrt{\sigma^{2}} \], Sample Standard Deviation - \[s = \sqrt{s^{2}} \]. The average of mean differences = [(3.25-1)2 + (3-3.25)2+ (4-3.25)2 + (5-3.25)2]/4 = 2.06. Step by step calculation:Follow these below steps using the above formulas to understand how to calculate standard deviation for the frequency table data setstep 1: find the mid-point for each group or range of the frequency table.step 2: calculate the number of samples of a data set by summing up the frequencies.step 3: find the mean for the grouped data by dividing the addition of multiplication of each group mid-point and frequency of the data set by the number of samples.step 4: calculate the variance for the frequency table data by using the above formula.step 5:estimate standard deviation for the frequency table by taking square root of the variance. A low Standard Deviation means that the value is close to the mean of the A low Standard Deviation means that the value is close to the mean of the set (also known as the expected value), and a high Standard Deviation means that the value is spread over a wider area. The standard deviation of a random variable is calculated by taking the square root of the product of the squared difference between the random variable, x, and the expected value () and the probability associated value of the random variable. Consider the following example. An example of this in industrial applications is quality control for Standard deviation formula is used to find the values of a particular data that is dispersed. Calculating Standard Deviation: A Step-by-Step Guide. The Standard Deviation, abbreviated as SD and represented by the letter ", indicates how far a value has varied from the mean value. is the standard deviation of the data set, N is the number of data inside the data set, X is each value of the data, and is the mean of the population. 6. Around 68% of scores are between 40 and 60. Lower the deviation, the close the numbers are dispersed from the mean. Find the exact standard deviation and mean. So we Correct answer: Explanation: The following is the formula for standard deviation: Here is a breakdown of what that formula is telling you to do: 1. In descriptive statistics, the standard deviation is the degree of dispersion or scatter of data points relative to the mean. If you want to cite this source, you can copy and paste the citation or click the Cite this Scribbr article button to automatically add the citation to our free Citation Generator. Solution: Problem 4: There were 30 students in the class. The probability distribution's standard deviation \[ X = x^{2}P(X = x) \]. Simple. Step 3: Find the mean of those squared deviations. The best measure of dispersion is standard deviation, which is why it is the most often used measure of dispersion: 2. The standard deviation of a set of data is equal to the square root of the variance. The mean (M) ratings are the same for each group its the value on the x-axis when the curve is at its peak. Now, substitute the obtained values in the formula. This mean is known as the expected value of the experiment denoted by . Around 99.7% of values are within 3 standard deviations of the mean. \(x_i\) is calculated as the midpoint of each class. Lastly, Standard Deviation is square-root of variance. There Are Two Types of Standard Deviation. Retrieved December 8, 2022, The standard deviation for continuous distribution is a measurement of its spread. Well use a small data set of 6 scores to walk through the steps. 1. Let us learn to calculate the standard deviation of grouped and ungrouped data and the standard deviation of a random variable. acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Full Stack Development with React & Node JS (Live), Fundamentals of Java Collection Framework, Full Stack Development with React & Node JS(Live), GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam. P(X=5)=P(1,4)+P(2,3)+P(3,2)+P(4,1) = 4/36 = 1/9, P(X=6)=P(1,5)+P(2,4)+P(3,3)+P(4,2)+P(5,1) = 5/36, P(X=7)=P(1,6)+P(2,5)+P(3,4)+P(4,3)+P(5,2)+P(6,1) = 6/36 = 1/6, P(X=8)=P(2,6)+P(3,5)+P(4,4)+P(5,3)+P(6,2) = 5/36, P(X=9)=P(3,6)+P(4,5)+P(5,4)+P(6,3) = 4/36 = 1/9, P(X=10)=P(4,6)+P(5,5)+P(6,4)= 3/36 = 1/12. The standard deviation of the probability distribution of X, = \(\sqrt{(x - )^2 P(X=x)}\), This is also equivalent to = \(\sqrt{E(X)^2-[E(X)]^2}\). The steps of calculating the standard deviation of discrete series with the direct method are as follows: Example: Calculate the standard deviation of the following series by Direct Method. Calculate the standard deviation of their heights. In order to arrive at the standard deviation of the set of numbers we have step wise process. We tend to know the average outcome when the difference between the theoretical probability of an event and its relative frequency approaches zero. The method of determining the deviation of a data point is used to calculate the degree of variance. The higher the CV, the higher the standard deviation relative to the mean. They have different representations and are calculated differently. What is the best measure of dispersion, and how? This is a function that gives each outcome in a sample space a numerical value. If we get a low standard deviation then it means that the values tend to be close to the mean whereas a high standard deviation tells us that the values are far from the mean value. In general, a CV value greater than 1 is often considered high. One of the most basic approaches of statistical analysis is the standard deviation. The last step is to calculate the standard deviation of the frequency distribution series using the formula. These arguments can then be stored in the variables such that arg[0] is the number of elements (n) or size of array followed by n elements as individual elements of the array. It is important to observe that the value of standard deviation can never be negative. Standard deviation is a measure used in calculating the dispersion of a data set. Arithmetic mean. When the difference between the theoretical probability of an event and its relative frequency get closer to each other, we tend to know the average outcome. What is the Difference between Interactive and Script Mode in Python Programming? Standard deviation is widely used in experimental and industrial settings to test models against real-world data. 4. It's one of a probability & statistics tools using the mid-point method to find the deviation of the grouped data. The methods used in this article are as follows: Standard Deviation in general terms can be explained as the divergence of the participants from the mean value among the group of values. The variance measures the average degree to which each point differs from the meanthe average of all data points. But, if we select another sample from the same population, it may obtain a different value. Let X represent the total of the numbers obtained by rolling two fair dice. The variance of a data set is the average square distance between the mean value and each data value, as previously stated. Find the standard deviation of their marks. As a result, we conclude that: is a good indicator of how dispersed or scattered something is. It is because standard deviation takes into account every value of a data set along with its algebraic signs. Followed by which we find the mean, variance and standard deviation as mentioned above. For example, the sum of uncorrelated distributions (random variables) also has a variance that is the sum of the variances of those distributions. Solve for the mean (average) of the five test scores. The Principal Director of Defense Pricing and Contracting issues class deviations when necessary to allow organizations to deviate from the FAR Around 95% of scores are between 30 and 70. Standard deviation is expressed in the same units as the original values (e.g., minutes or meters). In this problem, we will first require the total 2022. Java code To Calculate Standard Deviation In this article, we will brief in on all the possible ways to calculate standard deviation in Java Programming. Whats the difference between standard deviation and variance? This is given as \(s=\sqrt{\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}\). The measure of spread for the probability distribution of a random variable determines the degree to which the values differ from the expected value. Determine the variance and standard deviation of the random variable X with the following probability distribution: Assume that the mean of the random variable X is . 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