Standard Deviation

 Standard Deviation

What is Standard Deviation?

Standard deviation is the measurement of the dispersion of the data set from its mean value. It is always measured in arithmetic value. Standard deviation is always positive and is denoted by σ (sigma). Standard Deviation or SD is very accurate and is preferred from other measures of dispersion.

The standard deviation (SD) is calculated as The square root of variance by determining each data point's deviation relative to the arithmetic mean. In case the data points are far from the mean, it denotes a higher deviation within the set of data. Hence, it indicates more spread-out the data; the higher is the standard deviation. 

The formula to calculate Standard Deviation is:

where:

x(i) = value of the i’th point in the set of data

x(bar) = the mean-value of the set of data

n = the number of data-points in the set of data 

Properties of Standard Deviation

ü Standard Deviation is only used in measuring dispersion or spread around the mean value of the data set.

ü Standard Deviation is always in positive value.

ü It determines the dispersion or variation that exists from the average value.

ü Standard Deviation is a very sensitive outlier. Any single outlier can distort the picture of dispersion.

ü For the data set with an approximately same mean value, the higher the dispersion or spread, the greater is the Standard Deviation.

ü Standard Deviation is zero when the values of a particular data set are the same.

ü While analyzing the normally distributed data, SD is used in conjunction along with the mean to calculate the data intervals.

If  = mean, S = Standard Deviation, and x = Value in the Data set, then

around 68% of the Data is in the interval:-    - S < x < mean + S.

around 95% of the Data is in the interval:-   - 2S < x <mean + 2S.

around 99% of the Data is in the interval:-   - 3S < x < mean + 3S.

Standard Deviation Calculation

Before calculating the Standard Deviation, it is essential to underline the three types of data distribution. These are:

1. Individual series

A single column denoting the observation is available here.

Score

28

34

48

69

73

78

84

89

93

 

2. Discrete series

There are two columns that represent different data. One column shows the observation, while the other column is for frequency corresponding to the observation column.

Marks	30	40	50	60	70	80
Number of students	5	6	4	9	10	8

3. Frequency distribution

It has two columns, one representing the observations, and the other is corresponding frequency.

Here the observations are classified further into intervals or classes.

Age	20-30	30-40	40-50	50-60	60-70	70-80
Number of people	34	45	30	20	15	10

 

Sigma for individual series

The Standard deviation for individual series can be calculated by three methods; these are:

A direct method to calculate Standard Deviation

Use the formula ∑X/N to calculate the arithmetic mean. After this, we calculate the deviations of all the observations from the mean value using the formula D= X-mean.

Now, the deviations, x, are squared and summed. The resultant value is then divided by the total number of observations. The square root of the above-derived value = Standard Deviation

The formula is - σ = √[∑D²/N]

Here, D = deviation of an item that is relative to mean. It is calculated as D= X- mean.

N= Number of observations

Short-cut method

In this method, any random value is assumed to calculate deviation. It is believed that the assumed value is in the Middle of the Range of Values. The short cut method is derived using the formula;

σ = √[(∑D²/N) – (∑D/N)²]

 

Step-deviation method

It is a simple form of the short-cut method. Here, we select a common factor C, among the deviations. All the deviation values reduce when divided by C, simplifying the calculations. The formula is;

SD (σ)= √[(∑D’²/N) – (∑D’/N)²]  × C

D'= step-deviation of Observations relative to an Assumed mean. It is calculated as D'= (X-A)/C

C= Common Factor chosen.

 

Sigma for discrete series

There are two ways to calculate Standard Deviation in discrete series, theses are:

Direct method

We know that in the discrete series, another frequency column is added; the direct method formula to calculate Standard Deviation is:

Standard deviation (σ) = √(∑fD²)/N)

Short-cut method

Standard deviation (σ) = √[(∑fD²/N) – (∑fD/N)²]

 

Sigma for frequency distribution

There are three different methods that can be used to calculate standard deviation in frequency distribution series; these methods are:

Direct method

The direct method employed to derive standard deviation in a frequency distribution is very similar to the discrete series done above. The value of observation (when used) in the frequency distribution is the only difference between the two series. Here, the mid-value of the class is determined by dividing the sum' of the upper value of the class and the lower value of the class. The value thus derived is used for calculation. The formula is;

Standard Deviation (σ) = √(∑fD²)/N)

In the calculation, D = Deviation of an item that is relative to mean value and is calculated as,

D = Xi – Mean

f= frequencies corresponding to the Observations

N = The summation of the frequency.

 

Step-deviation Method

The step- deviation method is the short cut method to determine Standard Deviation. The formula is:

Standard Deviation (σ) = √[(∑fD’²/N) – (∑fD’/N)²]  ×  C

In the above calculation, D'= Step-Deviation of the observations relative to the assumed value. It is calculated as - D'= (Xi-A)/C

N = The Summation of Frequency.

C = Common Factor chosen

Did you know?

Without Standard Deviation, one can’t compare two sets of data effectively. Suppose there are two data sets having the same average, does that imply that the sets of data are exactly the same? No. Forex. the data sets - 199, 200, 201 and other 0, 200, 400 have the same 200 average, yet they have different standard deviations. Here, the first data has a small standard deviation (s=1) in comparison to the second set of data (s=200).

 

Fun facts about Standard Deviation

Try to find out the average ‘daily high temperature’ of two cities. One inland city while the other near the ocean. It is important to underline that the ranges of daily high temp. of the coastal city is less than the inland city. Both the cities can have the same average daily High temperature, but the standard deviation in the daily High temperature for a city near to ocean is less in comparison to the inland city.

 

FAQs

Q  What does SD or standard deviation indicates?

A  The standard deviation (SD) indicates the amount of variability on an average in your set of data. On average, it tells us how far each and every score is available from the mean value. The SD is the most accurate measurement in comparison to other measures of dispersion available and can never be negative. Standard deviation is denoted by the symbol Sigma or σ.

In normal distributions, a higher standard deviation implies that the values are further away from the mean. Similarly, a lower standard deviation means the values are clustered very close to the arithmetic mean value.

Q  What is the difference between the variance and the standard deviation?

A  The difference between the standard deviation and the variance is as follows:

Variance means the average squared deviations that are measured from the mean, whereas Standard Deviation is calculated as the Square root of this number. Although both the measurements indicate variability in the distribution, however, their units differ:

Ø The standard deviation (SD) is expressed as the same unit that is available in the original value (example - meters, grams or minutes)

Ø The variance is denoted in larger units in comparison, such as a square meter.

Although the units measured of variance are a little difficult to understand initially, the variance is significant in the statistical test. 

Ethylene glycol

 Ethylene glycol

Ethylene glycol is an 'organic compound' that is extremely toxic. It is a clear, odourless, viscous liquid that has a sweet taste. Ethylene glycol finds extensive use in the industrial and commercial sectors. Ethylene glycol meant the glycol which is derived from Ethylene.

The chemical has both antifreeze and coolant properties. Ethylene glycol keeps your vehicle running during the winter season by not letting the engine freeze. Similarly, during the summer season, it acts as a coolant that keeps your vehicle running by preventing the engine's overheating.

The boiling point of Ethylene glycol is 197°C, and the density of the organic compound is 1.11 g/cm³. Ethane-1, 2-diol is the IUPAC name of Ethylene glycol, and its melting point is -12.9 °C.

Structure of Ethylene glycol

Ethylene glycol is also known as Monoethylene glycol. It is produced when ethylene oxide chemically reacts with water. The chemical formula of Ethylene glycol is C₂H₆O_2. The chemical structure of Ethylene glycol is as follows:

What are the uses of Ethylene glycol?

Ethylene glycol is a viscous liquid that acts as a raw material in the manufacturing of:

l An ingredient for making automotive coolant and antifreeze products.

l They are extensively used in the manufacturing of polyester fibres.

l An ingredient in printing inks, hydraulic fluids, and paint solvents

l A reagent used in making explosives, synthetic waxes, and alkyd resins

l It is also used in manufacturing fibreglass for bathtubs, bowling balls, and jet skis.

l It is also used in making industrial coolants for ventilation, gas compressors, air-conditioning systems, heating, and ice-skating rinks.

 

What is the molar mass of Ethylene glycol?

The molar mass of any chemical compound is the sum total of the atomic mass of every element that forms it. Ethylene glycol is composed of 6 hydrogens, 2 carbon, and 2 oxygen atoms. So,

M (molar mass of Ethylene glycol) = 12( carbon atomic mass) x 2 + 1(hydrogen atomic mass) x 6 + 16(oxygen atomic mass) x 2 = 62.07(g/mol)

 

Did you know?

Ethylene glycol is too poisonous for human or pets consumption. It may lead to severe illness and, in extreme cases, even death. The tricky part is that ethylene glycol's taste is appealing to our pets like dogs and cats. A tablespoon of ethylene glycol is fatal for your dog, while one teaspoon is enough for death for a cat. If by accident, your pet has consumed the chemical, an antidote can be given immediately.

 

Properties of Ethylene Glycol – C2H6O2

The chemical formula of Ethylene glycol	C2H6O2
Molar mass/Molecular weight	62.07 g/mol
Melting point	-12.9 °C
Density	1.11 g/cm³
Boiling point	197.3 °C

 

How to produce Ethylene glycol?

There are different methods to produce Ethylene glycol; some of these are given below:

1) From Ethylene

 Hydroxylation of Ethylene produces Ethylene glycol. Ethylene, when treated with baeyer's reagent (Cold dilute alkaline- solution of Potassium Permanganate) leads to hydroxylation at both Carbon atoms.

In this way, Ethylene glycol is produced from Ethylene.

 

2) From Ethylene oxide

Hydrolysis of ethylene oxide produces Ethylene glycol. Ethylene oxide produced by ethylene's catalytic oxidation when hydrolyzed in the presence of acid at high temperature yields Ethylene oxide.

Fun facts

ü Ethylene glycol is miscible with water.

ü It is the simplest member of the glycol family.

ü It is slightly flammable.

ü It takes about ten days for Ethylene glycol to break down in the air.

ü It vigorously reacts with oxidizing agents.

ü Acrid smoke is emitted when heated to decompose. It emits irritating fumes that are hazardous.

 

FAQs

Q  Which products contain Ethylene glycol?

A  Ethylene glycol has many industrial and commercial applications. There are many household products that contain ethylene glycol. It includes:

Ø Antifreeze.

Ø Car wash fluids.

Ø De-icing products.

Ø Detergents.

Ø Vehicle brake fluids.

Ø Industrial solvents.

Ø Paints.

Ø Cosmetics.

 

Q  What is Ethylene glycol poisoning?

A  Drinking ethylene glycol is very harmful as it is a poisonous chemical substance. The feeling of drinking alcohol is the first symptom after consuming Ethylene glycol. Later other symptoms emerge that include nausea, convulsions, vomiting, unconsciousness, and in extreme cases, it may result in the death of the person. However, an antidote is available to save a person's life, but the antidote must be given immediately.

It is to be noted that skin contact with a smaller quantity of antifreeze solution will not affect your health, only when Ethylene glycol ingested in large amounts may be fatal.

 

Q  How is ethylene glycol harmful to humans?

A  When ethylene glycol enters the human body, it breaks down and forms chemicals. These chemicals later become crystals and start accumulating in different organs of the human body. It forms crystals in the kidney affecting the kidney function. Ethylene glycol creates acidic chemicals in the human body. These chemicals can change the body's acid/base balance, thus affecting the lungs, heart, and nervous system of the human body.