# Properties Of Variance And Standard Deviation In Statistics Pdf

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However, not every one of them is inhabited. Any finite number divided by infinity is as near nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely products of a deranged imagination.

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.

## 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 deviation

In statistics, the range is a measure of the total spread of values in a quantitative dataset. Unlike other more popular measures of dispersion, the range actually measures total dispersion between the smallest and largest values rather than relative dispersion around a measure of central tendency. The range is interpreted as t he overall dispersion of values in a dataset or, more literally, as the difference between the largest and the smallest value in a dataset. The range is measured in the same units as the variable of reference and, thus, has a direct interpretation as such. This can be useful when comparing similar variables but of little use when comparing variables measured in different units.

In statistics , the standard deviation is a measure of the amount of variation or dispersion of a set of values. The standard deviation of a random variable , sample , statistical population , data set , or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust , than the average absolute deviation. The standard deviation of a population or sample and the standard error of a statistic e. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and computing a mean for each sample. The mean's standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size. For example, a poll's standard error what is reported as the margin of error of the poll , is the expected standard deviation of the estimated mean if the same poll were to be conducted multiple times. The Standard Deviation is a measure of how spreads out the numbers are. The formula is easy: it is the square root of the Variance. The Variance is defined as: The average of the squared differences from the Mean. Then for each number: subtract the Mean and square the result (the squared difference).

## Unit 6: Topic 2: Properties of Variance & Standard Deviation

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Standard deviation and variance are types of statistical properties that measure dispersion around a central tendency, most commonly the arithmetic mean. They are descriptive statistics that measure variability around a mean for continuous data. The greater the standard deviation and variance of a particular set of scores, the more spread out the observations or data points are around the mean. Standard deviation and variance are closely related descriptive statistics, though standard deviation is more commonly used because it is more intuitive with respect to units of measurement; variance is reported in the squared values of units of measurement, whereas standard deviation is reported in the same units as the data. For example, to describe data on how long it took respondents to take Suggested ways of teaching this topic: Brainstorming and Guided Discovery. The teacher might start with the following brainstorming questions to revise the previous lesson.

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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. An important consequence of this is that the mean of any symmetric random variable continuous or discrete is always on the axis of symmetry of the distribution; for a continuous random variable, this means the axis of symmetry of the pdf. The module Discrete probability distributions gives formulas for the mean and variance of a linear transformation of a discrete random variable.

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