Standard Deviation Formula: The standard deviation, represented by the symbol 'σ', is the positive square root of the variance. It is a fundamental statistical tool used to measure the extent of data dispersion from the mean. Abbreviated as SD, it indicates how much individual data points deviate from the average value.
A low standard deviation suggests that the values are closely clustered around the mean, while a high standard deviation signifies that the values are more widely spread from the mean.
There exist distinct formulas for calculating the standard deviation of both grouped and ungrouped data. Additionally, various formulas cater to determining the standard deviation of a random variable. Let's explore each of these formulas in more detail.
What is Standard Deviation Formula?
Standard deviation represents the extent of dispersion or the spread of data points concerning their mean in descriptive statistics. It quantifies how values are distributed across a data sample and serves as a measure of the variation of data points from the mean. It is the square root of the variance of a dataset, sample, statistical population, random variable, or probability distribution. When considering n observations (x₁, x₂, ..., xₙ), the mean deviation of the values from the mean is calculated as Σᵢ₌₁ⁿ(xi - 𝑥̄)².
However, only summing the squares of deviations from the mean does not seem to be an appropriate measure of dispersion. If the average of squared differences from the mean is small, it suggests that the observations (xi) are close to the mean (𝑥̄), indicating lower dispersion. Conversely, if this sum is large, it implies a higher degree of dispersion of observations from the mean (𝑥̄).
Hence, Σᵢ₌₁ⁿ(xi - 𝑥̄)² serves as a reasonable indicator of the degree of dispersion or scatter.
We consider 1/n times the sum from i equals 1 to n of (xi - 𝑥̄)² as an appropriate measure of dispersion, termed the variance (σ²). The square root of the variance, when positive, is recognized as the standard deviation.
Standard Deviation Formula
Standard deviation quantifies the spread of statistical data by estimating the dispersion of data points. It is a measure of the degree of spread, commonly found within summary statistics, assessing the average square distance between each data value and the mean in a dataset. The standard deviation is pivotal in defining the range of data values around the mean. Two primary formulas are used to calculate the standard deviation of sample data and that of a given population.
It's important to note the similarity between both formulas, except for the difference in the denominator: 'N' for the population standard deviation and 'n-1' for the sample standard deviation. The sample mean is calculated using a subset of data values from the population, introducing some uncertainty or bias into the estimation of the population mean. To address this, the denominator of the sample standard deviation is adjusted to 'n-1' instead of 'n.' This correction, known as Bessel's correction, helps mitigate the bias in the calculation of standard deviation.
Formula for Computing Standard Deviation Two primary types of datasets are populations and samples. A population represents the entire group under study, while a sample is a smaller subset selected from the population. The formulas for calculating standard deviations for populations and samples exhibit slight differences. The population standard deviation formula is expressed as:
σ = √(1/N) ∑ᵢ₌₁ᴺ (Xi - μ)²
Here, σ = Symbol for population standard deviation μ = Population mean N = Total number of observations
Likewise, the formula for sample standard deviation is:
s = √(1/(n - 1)) ∑ᵢ₌₁ⁿ (xi - 𝑥̄)²
Here:
s = Symbol for sample standard deviation
𝑥̄ = Mean value of the observations
n = Total number of observations
Calculating Standard Deviation
When computing the population standard deviation, follow these steps to determine the standard deviation of a set of data values:
Determine the mean, which represents the arithmetic average of the observations.
Calculate the squared differences from the mean for each data value. (Square the difference between each data value and the mean: (Data value - mean)²)
Compute the average of the squared differences. (Variance = Sum of squared differences ÷ Number of observations)
Finally, find the square root of the variance.
(Standard deviation = √Variance)
Standard Deviation Formula for Individual Data Points
Calculating the standard deviation varies depending on the nature of the data. The deviation of data from its average position, or mean, can be determined using three distinct methods:
-
Actual mean method
-
Assumed mean method
-
Step deviation method
Standard Deviation Formula using the Actual Mean Approach
This method involves initially calculating the mean of the dataset (𝑥̄), then determining the deviations of each data point from this mean. Subsequently, the standard deviation is computed using the following formula by the actual mean method:
σ = √(∑(x - 𝑥̄)² / n)
, where n equals the total number of observations.
For instance, let's take the data observations: 3, 2, 5, 6. The mean of these data points is obtained as (3 + 2 + 5 + 6) / 4 = 16 / 4 = 4.
The sum of the squared differences from the mean equals (4-3)² + (2-4)² + (5-4)² + (6-4)² = 10.
Variance is computed as the sum of squared differences from the mean divided by the number of data points: 10 / 4 = 2.5.
Consequently, the standard deviation equals √2.5 ≈ 1.58.
Standard Deviation Formula via the Assumed Mean Approach
In instances where the x values are considerably large, selecting an arbitrary value (A) as the mean proves more practical, as computing the actual mean becomes challenging. The deviation from this chosen assumed mean is calculated as d = x - A. The formula for standard deviation by the assumed mean method is:
σ = √[(∑(d)² / n) - (∑d / n)²]
Standard Deviation Formula Using the Step Deviation Method
The standard deviation of grouped data can also be determined through the "step deviation method." In this approach, an arbitrary data value, A, is selected as the assumed mean. Subsequently, the deviations of all data values are calculated using d = x - A. The subsequent step involves computing the step deviations (d') as d' = d / i, where 'i' represents a common factor of all 'd' values (select any common factor if multiple factors exist).
The standard deviation of ungrouped data via the step deviation method is determined by the following formula:
σ = √[(∑(d')² / n) - (∑d' / n)²] × i,
where 'n' signifies the total number of data values.
Standard Deviation Formula of Grouped Data (Discrete)
When dealing with grouped data points, the initial step involves constructing a frequency distribution. Similar to ungrouped data, calculating the standard deviation of grouped data is achievable using three methods: the actual mean method, assumed mean method, and step deviation method. Let's explore each of these techniques in detail.
Standard Deviation Formula of Discrete Data using the Actual Mean Method
For 'n' observations, x₁, x₂, ..., xₙ, and their corresponding frequencies, f₁, f₂, f₃, ..., fₙ, the formula for standard deviation is:
σ = √(1/n) ∑ᵢ₌₁ⁿ fi(xi - 𝑥̄)²
. Here:
n = total frequency = ∑ᵢ₌₁ⁿ fi
𝑥̄ = mean
Standard Deviation Formula for Discrete Data Using the Assumed Mean Method
In situations where the data values are considerably large, one of these values is selected as the mean (termed the assumed mean, A). Subsequently, the deviation of each data value from the assumed mean is computed as d = x - A. The formula for standard deviation using the assumed mean method is:
σ = √[(∑(fd)² / n) - (∑fd / n)²]
, where:
'f' represents the frequency of the corresponding data value 'x', and
'n' stands for the total frequency.
Standard Deviation Formula for Discrete Data Using the Step Deviation Method
The standard deviation of grouped data can be determined through the "step deviation method." Similarly to other methods, an assumed mean (A) is chosen, and the deviations of data values are calculated using d = x - A. Within this approach, step deviations (d') are established, computed as d' = d / i, where 'i' represents a common factor among the values denoted by 'd' (multiple factors can exist, and any one may be selected).
The standard deviation using the step deviation method is calculated via the formula:
σ = √[(∑(fd')² / n) - (∑fd' / n)²] × i
,
where:
'f' signifies the frequency of data values
'n' denotes the total frequency
'i' represents a common factor among all 'd' values, where d = x - A (A = assumed mean)
d' = d / i
Standard Deviation for Grouped Data (Continuous) In the case of a continuous frequency distribution, each class is substituted with its midpoint. The standard deviation is computed using a similar technique to that used in discrete frequency distribution. For instance, consider the following example: x i is determined as the midpoint of each class, calculated by the formula: (lower bound + upper bound) / 2. Subsequently, the standard deviation formula is applied in the same manner.
Presently, the standard deviation can be computed using the grouped data formulas, using either the actual mean method, assumed mean method, or step deviation method.
Extent to which values deviate from the expected value. A random variable, denoted as X, Y, or Z, acts as a function assigning numerical values to each outcome in a sample space. For a random variable X, the standard deviation is computed by taking the square root of the sum of the product of the squared differences between the random variable X and the expected value (𝜇 or E(X)) and the associated probability of the random variable. The formula for the standard deviation of the probability distribution of X, 𝜎, is given by: σ= ∑[(x−μ) 2 ⋅P(x)]
An alternative method to find the standard deviation of random variables is through the following shortcuts:
σ= E(X 2 )−[E(X)] 2 or σ= ∑[x 2 ⋅P(x)]−μ 2
Standard Deviation Formula in Probability Distributions
Experimental probability results from numerous trials. As the variance between the theoretical probability of an event and its relative frequency diminishes, it indicates an approaching average outcome, known as the experiment's expected value, denoted by 𝜇.
In a normal distribution, the mean is zero and the standard deviation is 1.
For a binomial experiment where the number of successes serves as a random variable, the standard deviation is determined by: 𝜎 = √npq, with mean 𝜇 = np, where n represents the number of trials, p is the probability of success, and q (1 - p) represents the probability of failure.
In a Poisson distribution, the standard deviation is expressed as 𝜎 = √λt, where λ signifies the average number of successes within a time interval t.
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