# Properties of variance and standard deviation pdf

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- Unit 6: Topic 2: Properties of Variance & Standard Deviation
- Understanding and calculating variance

*Adapted from this comic from xkcd. We are currently in the process of editing Probability! If you see any typos, potential edits or changes in this Chapter, please note them here.*

The sampling distribution of a statistic is the distribution of the statistic for all possible samples from the same population of a given size. Suppose you randomly sampled 10 women between the ages of 21 and 35 years from the population of women in Houston, Texas, and then computed the mean height of your sample. You would not expect your sample mean to be equal to the mean of all women in Houston.

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Chapter 3 developed a general framework for modeling random outcomes and events. This framework can be applied to any set of random outcomes, no matter how complex. However, many of the random outcomes we are interested in are quantitative, that is, they can be described by a number. This chapter will develop these tools. A random variable is a number whose value depends on a random outcome. The idea here is that we are going to use a random variable to describe some but not necessarily every aspect of the outcome. All of these random variables are defined in terms of the underlying outcome, but we can also define random variables in terms of other random variables.

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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.

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. In this module, we will prove that the same formulas apply for continuous random variables.

The Standard Deviation is a measure of how spreads out the numbers are. Its symbol is σ (the greek letter sigma). The formula is easy: it is the square root.

## Unit 6: Topic 2: Properties of Variance & 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.

In probability theory and statistics , variance is the expectation of the squared deviation of a random variable from its mean. In other words, it measures how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics , statistical inference , hypothesis testing , goodness of fit , and Monte Carlo sampling.

*Published on September 24, by Pritha Bhandari. Revised on October 12, *

### Understanding and calculating variance

Suggested ways of teaching this topic: Brainstorming and Guided Discovery. The teacher might start with the following brainstorming questions to revise the previous lesson. The formula is easy: it is the square root of the Variance. Deviation just means how far from the normal. The Variance is defined as: The average of the squared differences from the Mean. To calculate the variance, we follow the following steps:.

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. In many cases, the population distribution is described by an idealized, continuous distribution function.

Some properties of the sample mean and variance of normal data are carefully explained. Pointing out these and other proper- ties in classrooms may have.

#### Variance vs standard deviation

In probability theory and statistics , the exponential distribution is the probability distribution of the time between events in a Poisson point process , i. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution , and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts. The exponential distribution is not the same as the class of exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution , binomial distribution , gamma distribution , Poisson , and many others. The probability density function pdf of an exponential distribution is.

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.

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