# Standard Deviation And Variance Example Pdf

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Published: 10.04.2021  Typical Analysis Procedure. Enter search terms or a module, class or function name. While the whole population of a group has certain characteristics, we can typically never measure all of them.

This means that over the long term of doing an experiment over and over, you would expect this average.

We have seen that for a discrete random variable, that the expected value is the sum of all xP x. For continuous random variables, P x is the probability density function, and integration takes the place of addition. Let f x be a probability density function on the domain [a,b] , then the expected value of f x is.

## Calculating Variance and Standard Deviation

Measures of central tendency mean, median and mode provide information on the data values at the centre of the data set. Measures of dispersion quartiles, percentiles, ranges provide information on the spread of the data around the centre. In this section we will look at two more measures of dispersion called the variance and the standard deviation.

The variance of the data is the average squared distance between the mean and each data value. It might seem strange that it is written in squared form, but you will see why soon when we discuss the standard deviation. It has squared units. For example, the variance of a set of heights measured in centimetres will be given in centimeters squared. Since the population variance is squared, it is not directly comparable with the mean or the data themselves.

In the next section we will describe a different measure of dispersion, the standard deviation, which has the same units as the data. Since the variance is a squared quantity, it cannot be directly compared to the data values or the mean value of a data set.

It is therefore more useful to have a quantity which is the square root of the variance. This quantity is known as the standard deviation. In statistics, the standard deviation is a very common measure of dispersion.

Standard deviation measures how spread out the values in a data set are around the mean. More precisely, it is a measure of the average distance between the values of the data in the set and the mean. If the data values are all similar, then the standard deviation will be low closer to zero.

If the data values are highly variable, then the standard variation is high further from zero. The standard deviation is always a positive number and is always measured in the same units as the original data. For example, if the data are distance measurements in kilogrammes, the standard deviation will also be measured in kilogrammes.

The mean and the standard deviation of a set of data are usually reported together. In a certain sense, the standard deviation is a natural measure of dispersion if the centre of the data is taken as the mean.

It is often useful to set your data out in a table so that you can apply the formulae easily. What is the variance and standard deviation of the possibilities associated with rolling a fair die? A large standard deviation indicates that the data values are far from the mean and a small standard deviation indicates that they are clustered closely around the mean.

The following figures show plots of the data sets with the mean and standard deviation indicated on each. You can see how the standard deviation is larger when the data are more spread out. The standard deviation may also be thought of as a measure of uncertainty. In the physical sciences, for example, the reported standard deviation of a group of repeated measurements represents the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is very important: if the mean of the measurements is too far away from the prediction with the distance measured in standard deviations , then we consider the measurements as contradicting the prediction.

This makes sense since they fall outside the range of values that could reasonably be expected to occur if the prediction were correct.

Siyavula Practice gives you access to unlimited questions with answers that help you learn. Practise anywhere, anytime, and on any device! Bridget surveyed the price of petrol at petrol stations in Cape Town and Durban.

The data, in rands per litre, are given below. The standard deviation of Cape Town's prices is lower than that of Durban's. That means that Cape Town has more consistent less variable prices than Durban. All times are in seconds. We are asked how many values are further than one standard deviation from the mean, meaning outside the interval. Video: 23CV. Video: 23CW. Do you need more Practice? Sign up to practise now.

Exercise Find the mean price in each city and then state which city has the lower mean. Durban has the lower mean. Which city has the more consistently priced petrol? Give reasons for your answer. Compute the mean and variance of the following set of values. How many of the athletes' times are more than one standard deviation away from the mean? ## Standard Error of the Mean vs. Standard Deviation: The Difference

Measures of central tendency mean, median and mode provide information on the data values at the centre of the data set. Measures of dispersion quartiles, percentiles, ranges provide information on the spread of the data around the centre. In this section we will look at two more measures of dispersion called the variance and the standard deviation. The variance of the data is the average squared distance between the mean and each data value. It might seem strange that it is written in squared form, but you will see why soon when we discuss the standard deviation.

Quantitative Methods 1 Reading 8. Probability Concepts Subject 6. Why should I choose AnalystNotes? AnalystNotes specializes in helping candidates pass. Find out more. Subject 6. Standard deviation is a measure of dispersion that determines how far data is spread from the mean. Standard deviation is found by calculating a different.

## 4.2 Mean or Expected Value and Standard Deviation

With discrete random variables, we often calculated the probability that a trial would result in a particular outcome. For example, we might calculate the probability that a roll of three dice would have a sum of 5. The situation is different for continuous random variables. For example, suppose we measure the length of time cars have to wait at an intersection for the green light. If the traffic light has a cycle lasting 30 seconds, then 8.

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When introducing the topic of random variables, we noted that the two types — discrete and continuous — require different approaches. The equivalent quantity for a continuous random variable, not surprisingly, involves an integral rather than a sum. Several of the points made when the mean was introduced for discrete random variables apply to the case of continuous random variables, with appropriate modification. Recall that mean is a measure of 'central location' of a random variable.

### Calculating Variance and Standard Deviation

Previous: 2. Next: 2. Analogous to the discrete case, we can define the expected value, variance, and standard deviation of a continuous random variable. These quantities have the same interpretation as in the discrete setting. The expectation of a random variable is a measure of the centre of the distribution, its mean value. The variance and standard deviation are measures of the horizontal spread or dispersion of the random variable. The following animation encapsulates the concepts of the CDF, PDF, expected value, and standard deviation of a normal random variable.

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To better describe the variation, we will introduce two other measures of variation​—variance and standard deviation. (the variance is the square of the standard.

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