Standard deviation (simply explained)

Summary

This video script explains the concept of standard deviation as a measure of data dispersion around the mean. It outlines the process of calculating standard deviation using two different formulas, depending on whether the data represents a sample or the entire population. The script also differentiates between standard deviation and variance, emphasizing the importance of using standard deviation for easier data interpretation. A tip is provided to calculate standard deviation using an online tool, Beta Tab, for convenience.

Takeaways

• 📏 Standard deviation is a measure of how much data scatters around the mean, indicating the variability within a dataset.
• 🧮 To calculate the mean, sum all individual values and divide by the number of individuals in the dataset.
• 📉 Deviations from the mean are calculated by subtracting the mean from each individual data point.
• 🔢 The standard deviation is found by taking the square root of the average of the squared deviations from the mean.
• 🌐 There are two formulas for standard deviation: one for a population (dividing by n) and one for a sample (dividing by n-1).
• 🔄 The use of n-1 in the sample formula provides an unbiased estimate of the population standard deviation.
• 📊 The difference between standard deviation and variance is that variance is the square of the standard deviation, without taking the square root.
• 🔗 The variance is the squared average distance from the mean, and it is the square of the standard deviation.
• 📊 Using standard deviation is recommended over variance for data interpretation as it retains the original data's unit of measurement.
• 🌐 An online tool like Beta Tab can be used to calculate standard deviation easily by inputting data into a table on the website.

Q & A

What is standard deviation?

Standard deviation is a measure of how much data scatters around the mean. It quantifies the average amount by which individual data points differ from the mean value.

How do you calculate the mean?

The mean is calculated by summing the heights (or any other data points) of all individuals and dividing it by the number of individuals.

What does the standard deviation tell us about the data?

The standard deviation tells us how much, on average, data points deviate from the mean, indicating the dispersion or spread of the data.

What is the formula for calculating standard deviation?

The formula for calculating standard deviation is the square root of the sum of the squared deviations of each data point from the mean, divided by the number of values (n) or n-1 (depending on whether it's a sample or the entire population).

Why are there two different formulas for standard deviation?

There are two formulas because one is used when you have the entire population (dividing by n) and the other is used when you have a sample of the population (dividing by n-1) to estimate the population standard deviation.

What is the difference between standard deviation and variance?

Variance is the squared average distance from the mean, while standard deviation is the square root of the variance. Essentially, variance is the squared standard deviation.

Why is standard deviation preferred over variance when describing data?

Standard deviation is preferred because it is in the same unit as the original data, making it easier to interpret and understand, whereas variance is in squared units which can be harder to interpret.

What is the relationship between the standard deviation and the original data units?

The standard deviation is always in the same unit as the original data, which helps in making the measure of dispersion directly comparable to the data.

How can one calculate standard deviation easily?

Standard deviation can be calculated easily using online tools like Beta Tab on datadept.net, where you simply copy your data into a table and select the variable to calculate.

What is the significance of the quadratic mean in standard deviation calculation?

The quadratic mean is significant because using the arithmetic mean would always result in zero deviation due to positive and negative deviations canceling each other out. The quadratic mean avoids this by squaring the deviations before averaging.

What tip does the video provide for calculating standard deviation?

The video suggests using online tools like Beta Tab on datadept.net for an easy calculation of standard deviation by simply entering the data into the provided table.

Outlines

00:00

📏 Understanding Standard Deviation

This paragraph introduces the concept of standard deviation as a measure of data dispersion around the mean. It explains how to calculate the mean, which is the sum of all individual heights divided by the number of individuals. The standard deviation quantifies how much each data point deviates from this mean, using an example where the mean height is 155 centimeters. The formula for standard deviation is presented, emphasizing the use of the square root of the sum of squared deviations divided by the number of values (n). The distinction between the arithmetic mean and the quadratic mean is highlighted, with the latter being essential for standard deviation calculations to avoid a zero result. The paragraph concludes with a note on the two formulas for standard deviation, one for the entire population (dividing by n) and one for samples (dividing by n-1), explaining the choice of formula based on whether the data represents a population or a sample.

05:02

🔍 Calculating Standard Deviation and Variance

The second paragraph delves into the difference between standard deviation and variance. It clarifies that while standard deviation measures the average distance of data points from the mean, variance measures the squared average distance. The relationship between the two is such that variance is the square of standard deviation, and vice versa. However, variance often results in a unit that does not align with the original data, making standard deviation, which retains the original unit, more interpretable and preferable for data description. The paragraph also provides a practical tip for calculating standard deviation using an online tool called Beta Tab, accessible via datadept.net, where users can input their data, select the variable, and easily obtain the standard deviation. The video ends with a farewell, inviting viewers to look forward to the next video.

Keywords

💡Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. In the video, it is described as a measure of how much data scatters around the mean. It is calculated by taking the square root of the variance, which is the average of the squared differences from the mean. The video uses the example of measuring the height of a group of people to illustrate how standard deviation indicates the average amount by which each person's height deviates from the mean height.

💡Mean

The mean, often referred to as the average, is calculated by summing all the values in a data set and then dividing by the number of values. In the context of the video, the mean is used as a reference point to determine how much each individual's data point deviates. For instance, the mean height of a group is calculated to understand how much each person's height differs from the average height of the group.

💡Data Scatter

Data scatter refers to the distribution of data points around a central value, such as the mean. The video explains that standard deviation is a measure of this scatter, indicating how spread out the data points are. A larger standard deviation indicates greater scatter, while a smaller one indicates that the data points are closer to the mean.

💡Variance

Variance is a statistical measure that represents the average of the squared differences from the mean. It is used to quantify the dispersion of a set of data points. The video clarifies that variance is the squared form of standard deviation, and it is the average distance from the mean, but squared. The variance is less intuitive than standard deviation because it is in squared units, making standard deviation the preferred measure for data interpretation.

💡Sample

A sample is a subset of a larger population that is used to represent and make inferences about the whole population. The video discusses that when calculating standard deviation for a sample rather than the entire population, a different formula is used (n-1 in the denominator instead of n). This adjustment accounts for the fact that samples may not perfectly represent the population, and it helps to provide a better estimate of the population's standard deviation.

💡Population

In statistics, a population refers to the entire set of individuals or items of interest that are the subject of a study. The video mentions that if one had data for an entire population, such as the height of all American professional soccer players, a specific formula for standard deviation would be used (dividing by n). This is in contrast to using a sample to estimate population statistics.

💡Arithmetic Mean

The arithmetic mean is the sum of all values in a data set divided by the number of values. The video points out that if the arithmetic mean were used to calculate deviations for standard deviation, the result would always be zero because positive and negative deviations would cancel each other out. This is why the quadratic mean is used instead.

💡Quadratic Mean

The quadratic mean is the square root of the arithmetic mean of the squares of the data points. It is used in the calculation of standard deviation to ensure that all deviations (whether positive or negative) contribute to the measure of dispersion. The video explains that using the quadratic mean prevents the issue of deviations canceling each other out, which would happen if the arithmetic mean were used.

💡Sigma (σ)

In the video, sigma (σ) represents the standard deviation in the formula for calculating it. It is used to denote the symbol for standard deviation in mathematical expressions. The formula presented in the video involves summing the squared deviations from the mean, dividing by the number of values (n or n-1 depending on whether it's a population or sample), and then taking the square root, which is represented by sigma.

💡Beta Tab

Beta Tab is mentioned in the video as a tool for calculating standard deviation. It is an online resource where users can input their data, select the variable they wish to analyze, and easily obtain the standard deviation. This tool is highlighted as a practical way for viewers to apply the concepts discussed in the video to real-world data sets.

Highlights

Standard deviation measures how much data scatters around the mean.

Calculating the mean involves summing all data points and dividing by their count.

Standard deviation quantifies the average deviation from the mean.

The formula for standard deviation involves squaring the deviations and taking the square root.

The arithmetic mean cannot be used for standard deviation calculation as it would always result in zero.

There are two formulas for standard deviation: one for population and one for sample data.

The population standard deviation is calculated using the formula with division by n.

The sample standard deviation uses the formula with division by n minus one to estimate the population standard deviation.

Variance is the squared average distance from the mean, as opposed to the standard deviation.

Variance is the square of the standard deviation, and its unit does not correspond to the original data.

Standard deviation is preferred over variance for data interpretation due to its ease of understanding and same unit as the original data.

A tip is provided for calculating standard deviation using an online tool called Beta Tab.

Datadept.net is recommended for easily calculating standard deviation by inputting data into a table.

The video concludes with a reminder to use standard deviation for sample data interpretation.

The presenter invites viewers to watch more videos and says goodbye.

Transcripts

today is about standard deviation after

this video you will know what standard

deviation is how you can calculate it

and why there are two different formulas

and finally what is the difference to

the variance

at the end of this video i have a tip

for you so let's get started

so what is the standard deviation the

standard deviation is a measure of how

much your data scatters around the mean

so the standard deviation has something

to do with the scatter of your data for

example how different the answers of

your respondents are

here's an example

let's say you measure the height of a

small group of people

the standard deviation tells us how much

your data scatters around the mean

so we first need to calculate the mean

you can get a mean simply by summing the

heights of all individuals and dividing

it by the number of individuals

let's say we get a mean value of

155 centimeters

now we want to know how much each person

deviates from the mean

so we look at the first person who

deviates 18 centimeters from the mean

value the second person deviates 8

centimeters from the mean value

and so on

finally person number six deviates six

centimeters from the mean value

so simply said people that are very

small or very tall deviate more from the

mean value

now of course you're not interested in

the deviation of each individual person

from the mean value

but you want to know how much the

persons deviate from the mean value on

average

so how much do these persons on average

deviate from the mean value this is what

the standard deviation tells us

in our example the average deviation

from the mean value is

12.06 centimeters and now of course the

next question is how can we calculate

the standard deviation you can calculate

the standard deviation with the

following formula

sigma is the standard deviation

n is the number of persons

x i is the size of one single person and

x dash is the mean value of all people

so the standard deviation is the root of

the sum of square deviations divided by

the number of values

for our example this means that we

calculate the size of the first person

minus the mean and square that then the

size of the second person minus the mean

and then square that and so on until we

arrive at the last person

then we divide this number by the number

of people so 6 and take the root of it

the result is then

12.06 centimeters

so each individual person has some

deviation from the mean

but on average the people deviate 12.06

centimeters from the mean

which is now our standard deviation

now you might notice one thing i always

talk about the average deviation from

the mean

but for the average deviation i would

actually just add up all deviations and

divide it by the number of participants

just like you calculate a mean value

right

you're absolutely right but there are

different mean values

in the case of the standard deviation

it's not the arithmetic mean which is

used

but the quadratic mean

if the arithmetic mean would be used the

result would be zero every time

so far so good but now there's one more

thing to consider

there are two slightly different

formulas for the standard deviation in

the first formula there is a deviation

by n and in the other one there is a

deviation by n minus one

but why that why are there two different

formulas

usually you want to know the standard

deviation of the whole population for

example you want to know the standard

deviation of hate of all american

professional soccer players

now if you had the hate of all american

soccer players

you would take this equation with one

divided by n

however it is usually not possible to

investigate the entire population

so you take a sample

then you use this sample to estimate the

standard deviation of the population

in that case you use this formula

therefore whenever you have data of the

whole population and you want to

calculate the standard deviation for

just this data you use 1 divided by n

therefore

whenever you have data of the whole

population and you want to calculate the

standard deviation for just this data

you use 1 divided by n

if you only have one sample and you want

to estimate the standard deviation you

use n minus 1.

so to keep it simple if your survey

doesn't cover the whole population you

always use the formula on the right side

likewise if you have conducted a

clinical study for example

then you also use the formula on the

right side to infer the population

let's look at the next question now

what is the difference between the

standard deviation and the variance

as you now know the standard deviation

is the average distance from the mean

the variance now is the squared average

distance from the mean

so we have one and the same formula the

only difference is that in order to

calculate the standard deviation we take

the root

in order to calculate the variance we

don't do that to put it the other way

around the variance is the squared

standard deviation and the standard

deviation is the root of the variance

however this squaring results in a

figure which is quite difficult to

interpret

since the unit of the calculated

variance does not correspond to the

original data

for this reason it is advisable to

always use the standard deviation to

describe a sample as this makes

interpretation a lot easier for you

the standard deviation is always in the

same unit as the original data

in our example this would be centimeters

and finally as promised i have a tip for

you

if you want to calculate the standard

deviation you can easily do it online

with beta tab

just visit datadept.net

copy your data into the table

select the variable you want to

calculate

and afterwards you will get the standard

deviation in a very easy way

i hope you enjoyed the video and see you

next time bye bye

you