Wednesday, January 25, 2023

What Does Standard Deviation Mean In Math

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How To Calculate Standard Error

Standard Deviation – Explained and Visualized

Standard error can be calculated using the formula below, where represents standard deviation and n represents sample size.

Standard error increases when standard deviation, i.e. the variance of the population, increases. Standard error decreases when sample size increases as the sample size gets closer to the true size of the population, the sample means cluster more and more around the true population mean.

Examples Of Standard Deviation

Unless youre sitting in a statistics class, you may think that standard deviation doesnt affect your everyday life. But youd be wrong! Even though most statisticians calculate standard deviation with computer programs and spreadsheets, its helpful to know how to do it by hand.

Here are some examples of situations that demonstrate how standard deviation is used.

Uncorrected Sample Standard Deviation

The formula for the population standard deviation can be applied to the sample, using the size of the sample as the size of the population . This estimator, denoted by sN, is known as the uncorrected sample standard deviation, or sometimes the standard deviation of the sample , and is defined as follows:


,\,x_,\,\ldots ,\,x_\}} are the observed values of the sample items, and x ¯ }} is the mean value of these observations, while the denominator N stands for the size of the sample: this is the square root of the sample variance, which is the average of the squared deviations about the sample mean.

This is a consistent estimator , and is the maximum-likelihood estimate when the population is normally distributed. However, this is a biased estimator, as the estimates are generally too low. The bias decreases as sample size grows, dropping off as 1/N, and thus is most significant for small or moderate sample sizes for N 75 the bias is below 1%. Thus for very large sample sizes, the uncorrected sample standard deviation is generally acceptable. This estimator also has a uniformly smaller mean squared error than the corrected sample standard deviation.

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Example : Tips Dataset

Get Tips Data

Let’s get the tips dataset from the seaborn library and assign it to the DataFrame df_tips.


Each row represents a unique meal at a restaurant for a party of people the dataset contains the following fields:

column name
financial amount of meal in U.S. dollars
tip financial amount of the meal’s tip in U.S. dollars
boolean to represent if server smokes or not
meal name
size count of people eating meal

Preview the first 5 rows of df_tips.


How to Calculate Standard Deviation: The Hard Way on Tips Dataset

The value for standard deviation is the square root of the variance. So first, let’s calculate variance. We can calculate the variance in the first three steps and the standard deviation in the fourth.

1) Calculate the mean

2) For each value, subtract the mean and square the result

3) Calculate the average of those squared differences

4) Calculate the square root of the variance

Let’s calculate the standard deviation of our total_bill column in df_tips.

1: Calculate the Mean

Use the mean method in pandas to calculate the mean of the total_bill column in df_tips.

2: Calculate the Squared Differences

Create a new column in df_tips that’s the difference between each total_bill value and mean_total_bill.


Preview the first few rows of the columns total_bill and total_bill_diff_from_mean.

3: Calculate the Variance

A Practical Example: Your Company Packages Sugar In 1 Kg Bags

Standard deviation (3)

When you weigh a sample of bags you get these results:

  • 1007g, 1032g, 1002g, 983g, 1004g, …
  • Mean = 1010g

Some values are less than 1000g … can you fix that?

The normal distribution of your measurements looks like this:

31% of the bags are less than 1000g,which is cheating the customer!

It is a random thing, so we can’t stop bags having less than 1000g, but we can try to reduce it a lot.

Let’s adjust the machine so that 1000g is:

  • at â3 standard deviations:
  • From the big bell curve above we see that 0.1% are less. But maybe that is too small.

  • at â2.5 standard deviations:
  • Below 3 is 0.1% and between 3 and 2.5 standard deviations is 0.5%, together that is 0.1% + 0.5% = 0.6%

So let us adjust the machine to have 1000g at â2.5 standard deviations from the mean.

Now, we can adjust it to:

  • increase the amount of sugar in each bag , or
  • make it more accurate

Let us try both.

Adjust the mean amount in each bag

The standard deviation is 20g, and we need 2.5 of them:

2.5 Ã 20g = 50g

So the machine should average 1050g, like this:

Adjust the accuracy of the machine

Or we can keep the same mean , but then we need 2.5 standard deviations to be equal to 10g:

10g / 2.5 = 4g

So the standard deviation should be 4g, like this:

Or perhaps we could have some combination of better accuracy and slightly larger average size, I will leave that up to you!

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Mean And Standard Deviation Formula

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 is represented by the symbol \. In Mathematical terms, sample mean formula is given as:

\= 1/n \

In the above sample mean formula

N is the sample size and

X is the correspond observed values

Standard Deviation – On the other hand, standard deviation perceives the significant amount of dispersion of observations when comes up close with data. In Mathematical terms, standard dev formula is given as:

Standard Deviation, = \

What Percentage Is 2 Standard Deviations From The Mean

For a data set that follows a normal distribution, approximately 95% of values will be within 2 standard deviations from the mean.

So, for every 1000 data points in the set, 950 will fall within the interval .

Going back to our example above, if the sample size is 1000, then we would expect 950 values to fall within the range .

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What Is 2 Standard Deviations From The Mean

When we say 2 standard deviations from the mean, we are talking about the following range of values:

where M is the mean of the data set and S is the standard deviation.

We know that any data value within this interval is at most 2 standard deviations from the mean. Some of this data is close to the mean, but a value 2 standard deviations above or below the mean is somewhat far away.

For example, if we have a data set with mean 200 and standard deviation 30 , then the interval

Is the range of values that are 2 standard deviations from the mean.

Example: In That Same School One Of Your Friends Is 185m Tall

Mean, Standard Deviation & Variance [IB Math AA SL/HL]

You can see on the bell curve that 1.85m is 3 standard deviations from the mean of 1.4, so:

Your friend’s height has a “z-score” of 3.0

It is also possible to calculate how many standard deviations 1.85 is from the mean

How far is 1.85 from the mean?

It is 1.85 – 1.4 = 0.45m from the mean

How many standard deviations is that? The standard deviation is 0.15m, so:

0.45m / 0.15m = 3 standard deviations

So to convert a value to a Standard Score :

  • first subtract the mean,
  • then divide by the Standard Deviation

And doing that is called “Standardizing”:

We can take any Normal Distribution and convert it to The Standard Normal Distribution.

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What Is Standard Deviation

Standard Deviation is a measure which shows how much variation from the mean exists. The standard deviation indicates a typical deviation from the mean. It is a popular measure of variability because it returns to the original units of measure of the data set. Like the variance, if the data points are close to the mean, there is a small variation whereas the data points are highly spread out from the mean, then it has a high variance. Standard deviation calculates the extent to which the values differ from the average. Standard Deviation, the most widely used measure of dispersion, is based on all values. Therefore a change in even one value affects the value of standard deviation. It is independent of origin but not of scale. It is also useful in certain advanced statistical problems.

Does Sample Size Affect Standard Deviation

Sample size does affect standard deviation. The sample size, N, appears in the denominator under the radical in the formula for standard deviation.

So, changing the value of N affects the standard deviation. Changing N also affects the mean.

Example 1: Changing N Changes Standard Deviation

For the data set S = , we have the following:

  • N = 3
  • Mean = 3 / 3 = 3)
  • Standard Deviation = 2

If we change the sample size by removing the third data point , we have:

  • S =
  • N = 2
  • Mean = 2 / 2 = 2)
  • Standard Deviation = 1.41421

So, changing N changed both the mean and standard deviation.

Of course, it is possible by chance that changing the sample size will leave the standard deviation unchanged.

Example 2: Changing N Leaves Standard Deviation Unchanged

For the data set S = , we have the following:

  • N = 3
  • Mean = 1.78868 / 3 = 3)
  • Standard Deviation = 0.70711

If we change the sample size by removing the third data point , we have:

  • S =
  • N = 2
  • Mean = 1.5 / 2 = 1.5)
  • Standard Deviation = 0.70711

So, changing N lead to a change in the mean, but leaves the standard deviation the same.

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Then Work Out The Mean Of Those Squared Differences

To work out the mean, add up all the values then divide by how many.

First add up all the values from the previous step.

But how do we say “add them all up” in mathematics? We use “Sigma”:

The handy Sigma Notation says to sum up as many terms as we want:

Sigma Notation

We want to add up all the values from 1 to N, where N=20 in our case because there are 20 values:

What Is Standard Deviation In Math Terms

Standard deviation (3)

A standard deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. The standard deviation is calculated as the square root of variance by determining each data points deviation relative to the mean.

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What Is Variance And Standard Deviation

Variance – The variance is a numerical value that represents how broadly individuals in a group may change. The variance will be larger if the individual observations change largely from the group mean and vice versa.

It is important to notice similarities between the variance of sample and variance population. They have different representations and are calculated differently. The variance of a population is represented by ² whereas the variance of a sample is represented by s².

Standard Deviation – Standard deviation is a measure of dispersion in statistics. It gives an estimation how individuals in data are dispersed from the mean value. Standard deviation is defined as the square root of the mean of a square of the deviation of all the values of a series derived from the arithmetic mean. It is also known as root mean square deviation.The symbol used to represent standard deviation is Greek Letter sigma .

Give An Example Of Standard Deviation

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. Consider data points 1, 3, 4, 5. The mean is 13/4 = 3.25. The average of mean differences = /4 = 2.06. The standard deviation = 2.06 = 1.43

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Why Is Standard Deviation A Useful Measure Of Variability

Although there are simpler ways to calculate variability, the standard deviation formula weighs unevenly spread out samples more than evenly spread samples. A higher standard deviation tells you that the distribution is not only more spread out, but also more unevenly spread out.

This means it gives you a better idea of your datas variability than simpler measures, such as the mean absolute deviation .

The MAD is similar to standard deviation but easier to calculate. First, you express each deviation from the mean in absolute values by converting them into positive numbers . Then, you calculate the mean of these absolute deviations.

Unlike the standard deviation, you dont have to calculate squares or square roots of numbers for the MAD. However, for that reason, it gives you a less precise measure of variability.

Lets take two samples with the same central tendency but different amounts of variability. Sample B is more variable than Sample A.


Standard Deviation Of Average Height For Adult Men

What is Standard Deviation and Mean Absolute Deviation | Math, Statistics for data science, ML

If the population of interest is approximately normally distributed, the standard deviation provides information on the proportion of observations above or below certain values. For example, the average height for adult men in the United States is about 70 inches , with a standard deviation of around 3 inches . This means that most men have a height within 3 inches of the mean ) one standard deviation and almost all men have a height within 6 inches of the mean ) two standard deviations. If the standard deviation were zero, then all men would be exactly 70 inches tall. If the standard deviation were 20 inches , then men would have much more variable heights, with a typical range of about 5090 inches . Three standard deviations account for 99.7% of the sample population being studied, assuming the distribution is normal or bell-shaped .

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Bounds On Standard Deviation

For a set of N> 4 data spanning a range of values R, an upper bound on the standard deviation s is given by s = 0.6R.An estimate of the standard deviation for N> 100 data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values R represents four standard deviations so that s R/4. This so-called range rule is useful in sample size estimation, as the range of possible values is easier to estimate than the standard deviation. Other divisors K of the range such that s R/K are available for other values of N and for non-normal distributions.

Does Multiplication Affect Standard Deviation

Multiplication affects standard deviation by a scaling factor. If we multiply every data point by a constant K, then the standard deviation is multiplied by the same factor K.

In fact, the mean is also scaled by the same factor K.

Example: Multiplication Scales Standard Deviation By A Factor Of K

For the data set S = , we have the following:

  • N = 3
  • Mean = 2 / 3 = 2)
  • Standard Deviation = 1

If we use multiplication by a factor of K = 4 on every point in the data set, we have:

  • S =
  • N = 3
  • Mean = 8 / 3 = 8)
  • Standard Deviation = 4

So, multiplying by K = 4 also multiplied the mean by 4 and multiplied standard deviation by 4 .

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How Is Standard Deviation Calculated

The formula for standard deviation makes use of three variables. The first variable is the value of each point within a data set, with a sum-number indicating each additional variable . The mean is applied to the values of the variable M and the number of data that is assigned to the variable n. Variance is the average of the values of squared differences from the arithmetic mean.

To calculate the mean value, the values of the data elements have to be added together and the total is divided by the number of data entities that were involved.

Standard deviation, denoted by the symbol , describes the square root of the mean of the squares of all the values of a series derived from the arithmetic mean which is also called the root-mean-square deviation. 0 is the smallest value of standard deviation since it cannot be negative. When the elements in a series are more isolated from the mean, then the standard deviation is also large.

The statistical tool of standard deviation is the measures of dispersion that computes the erraticism of the dispersion among the data. For instance, mean, median and mode are the measures of central tendency. Therefore, these are considered to be the central first order averages. The measures of dispersion that are mentioned directly over are averages of deviations that result from the average values, therefore these are called second-order averages.

Corrected Sample Standard Deviation

Standard deviation (3)

If the biased sample variance is used to compute an estimate of the population’s standard deviation, the result is

2 , }=}\gamma _}}\sum _^\left^}},}

where 2 denotes the population excess kurtosis. The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data.

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But There Is A Small Change With Sample Data

Our example has been for a Population .

But if the data is a Sample , then the calculation changes!

When you have “N” data values that are:

  • The Population: divide by N when calculating Variance
  • A Sample: divide by N-1 when calculating Variance

All other calculations stay the same, including how we calculated the mean.

Example: if our 5 dogs are just a sample of a bigger population of dogs, we divide by 4 instead of 5 like this:


Think of it as a “correction” when your data is only a sample.

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