Variance

Variance (σ2) in statistics is a measurement of the spread between numbers in a data set. That is, it measures how far each number in the set is from the mean and therefore from every other number in the set.

In investing, the variance of the returns among assets in a portfolio is analyzed as a means of achieving the best asset allocation. The variance equation, in financial terms, is a formula for comparing the performance of the Elements of a portfolio against each other and against the mean.

Variance is calculated by taking the differences between each number in the data set and the mean, then squaring the differences to make them positive, and finally dividing the sum of the squares by the number of values in the data set.

Variance is one of the key parameters in asset allocation, along with correlation. Calculating the variance of asset returns helps investors to develop better portfolios by optimizing the return-volatility trade-off in each of their investments.

Variance measures variability from the Average or mean. To investors, variability is volatility, and volatility is a measure of risk. Therefore, the variance statistic can help determine the risk an investor assumes when purchasing a specific security.

A large variance indicates that numbers in the set are far from the mean and from each other, while a small variance indicates the opposite. Variance can be negative. A variance value of zero indicates that all values within a set of numbers are identical.

Advantages and Disadvantages of Variance

Statisticians use variance to see how individual numbers relate to each other within a data set, rather than using broader mathematical techniques such as arranging numbers into quartiles. One drawback to variance is that it gives added weight to outliers, the numbers that are far from the mean. Squaring these numbers can skew the data.

The advantage of variance is that it treats all deviations from the mean the same regardless of their direction. The squared deviations cannot sum to zero and give the appearance of no variability at all in the data.

The drawback of variance is that it is not easily interpreted. Users of variance often employ it primarily in order to take the square root of its value, which indicates the standard deviation of the data set.

Types of Sampling

Sampling is defined as the process of selecting certain members or a subset of the Population to make statistical inferences from them and to estimate characteristics of the whole population. Sampling is widely used by researchers in market research so that they do not need to research the entire population to collect actionable insights.

Types of sampling

Probability Sampling

Probability sampling s a sampling method that selects random members of a population by setting a few selection criteria. These selection parameters allow every member to have the equal opportunities to be a part of various samples.

Probability Sampling is a sampling technique in which sample from a larger population are chosen using a method based on the theory of probability. This sampling method considers every member of the population and forms samples on the basis of a fixed process. For example, in a population of 1000 members, each of these members will have 1/1000 chances of being selected to be a part of a sample. It gets rid of bias in the population and gives a fair chance to all members to be included in the sample.

There are 4 types of probability sampling technique

Simple Random Sampling

One of the best probability sampling techniques that helps in saving time and Resources, is the Simple Random Sampling method. It is a trustworthy method of obtaining information where every single member of a population is chosen randomly, merely by chance and each individual has the exact same probability of being chosen to be a part of a sample.

Cluster Sampling

Cluster sampling is a method where the researchers divide the entire population into sections or clusters that represent a population. Clusters are identified and included in a sample on the basis of defining demographic parameters such as age, location, sex etc. which makes it extremely easy for a survey creator to derive effective inference from the feedback.

Systematic Sampling

Using systematic sampling method, members of a sample are chosen at regular intervals of a population. It requires selection of a starting point for the sample and sample size that can be repeated at regular intervals. This type of sampling method has a predefined interval and hence this sampling technique is the least time-consuming.

Stratified Random Sampling

Stratified Random sampling is a method where the population can be divided into smaller groups, that don’t overlap but represent the entire population together. While sampling, these groups can be organized and then draw a sample from each group separately.

Non-probability Sampling

Non probability sampling method is reliant on a researcher’s ability to select members at random. This sampling method is not a fixed or pre-defined selection process which makes it difficult for all elements of a population to have equal opportunities to be included in a sample.

Variance is a measure of how spread out numbers are in a data set. It is calculated by taking the square of the standard deviation.

The population variance is the variance of all the values in a population. The sample variance is the variance of a sample of values from a population.

The variance of a sum of random variables is the sum of the variances of the individual random variables. The variance of a product of random variables is the product of the variances of the individual random variables. The variance of a quotient of random variables is the quotient of the variances of the individual random variables.

The variance of a linear combination of random variables is the sum of the products of the coefficients of the random variables and the variances of the individual random variables.

The variance of a chi-squared distribution is equal to the degrees of freedom of the distribution. The variance of a normal distribution is equal to one. The variance of a t-distribution is equal to the degrees of freedom of the distribution divided by the square of the degrees of freedom plus one. The variance of an F-distribution is equal to the numerator degrees of freedom divided by the denominator degrees of freedom.

Variance is a useful measure of variability because it is easy to calculate and interpret. It is also relatively insensitive to outliers, which are data points that are far away from the rest of the data.

However, variance can be misleading if the data are not normally distributed. In this case, the Median may be a better measure of central tendency than the mean, and the interquartile range may be a better measure of variability than the variance.

Variance is also a useful measure of the spread of data in hypothesis testing. For example, the F-test compares the variances of two groups of data to see if they are significantly different. The t-test compares the means of two groups of data to see if they are significantly different.

In conclusion, variance is a useful measure of variability that can be used to describe the spread of data. It is easy to calculate and interpret, and it is relatively insensitive to outliers. However, variance can be misleading if the data are not normally distributed. In this case, the median may be a better measure of central tendency than the mean, and the interquartile range may be a better measure of variability than the variance. Variance is also a useful measure of the spread of data in hypothesis testing.

Here are some examples of how variance is used in statistics:

In regression analysis, variance is used to measure the spread of the residuals. The residuals are the differences between the observed values and the predicted values. The variance of the residuals is used to assess the goodness of fit of the regression model.
In ANOVA, variance is used to measure the variability between groups. The variability between groups is compared to the variability within groups to determine if there is a significant difference between the groups.
In hypothesis testing, variance is used to calculate the p-value. The p-value is the probability of obtaining the observed results if the null hypothesis is true. A low p-value indicates that the null hypothesis is unlikely to be true.

Variance is a powerful tool that can be used to understand and analyze data. It is important to understand how variance is calculated and how it is used in statistics.

What is a standard deviation?

The standard deviation is a measure of how spread out numbers are in a data set. A low standard deviation indicates that the data points tend to be very close to the mean, while a high standard deviation indicates that the data points are spread out over a large range of values.

How is standard deviation calculated?

The standard deviation is calculated by taking the square root of the variance. The variance is a measure of how much the data points vary from the mean. To calculate the variance, you first subtract the mean from each data point, then square the result, and then add up all the squared values. Finally, you divide the sum by the number of data points minus 1.

What is the difference between standard deviation and variance?

The standard deviation is a measure of how spread out numbers are in a data set, while the variance is a measure of how much the data points vary from the mean. The standard deviation is always positive, while the variance can be positive or negative.

What is the relationship between standard deviation and mean?

The standard deviation is inversely proportional to the mean. This means that as the mean increases, the standard deviation decreases, and vice versa.

What is the relationship between standard deviation and skewness?

The standard deviation is not related to the skewness of a data set. Skewness is a measure of how symmetrical a data set is. A data set with a positive skew is skewed to the right, while a data set with a negative skew is skewed to the left.

What is the relationship between standard deviation and kurtosis?

The standard deviation is not related to the kurtosis of a data set. Kurtosis is a measure of how peaked a data set is. A data set with a high kurtosis is more peaked than a data set with a low kurtosis.

What is the interquartile range?

The interquartile range (IQR) is a measure of variability that is resistant to outliers. It is calculated by finding the difference between the third and first quartiles.

What is the range?

The range is the difference between the largest and smallest values in a data set. It is a simple measure of variability, but it is not very resistant to outliers.

What is the coefficient of variation?

The coefficient of variation is a measure of how spread out a data set is relative to its mean. It is calculated by dividing the standard deviation by the mean.

What is the 68-95-99.7 rule?

The 68-95-99.7 rule is a rule of thumb that states that approximately 68% of the data points in a normal distribution will fall within 1 standard deviation of the mean, approximately 95% of the data points will fall within 2 standard deviations of the mean, and approximately 99.7% of the data points will fall within 3 standard deviations of the mean.

What is a normal distribution?

A normal distribution is a probability distribution that is bell-shaped. The mean, median, and mode of a normal distribution are all equal.

What is a z-score?

A z-score is a standardized measure of how far a data point is from the mean. It is calculated by subtracting the mean from the data point and then dividing the result by the standard deviation.

What is a confidence interval?

A confidence interval is a range of values that is likely to contain the true value of a population parameter. The confidence level is the probability that the confidence interval contains the true value of the population parameter.

What is a p-value?

A p-value is a measure of the evidence against the null hypothesis. A low p-value indicates that there is strong evidence against the null hypothesis.

What is a hypothesis test?

A hypothesis test is a statistical procedure that is used to test a claim about a population parameter. The null hypothesis is the claim that is being tested, and the alternative hypothesis is the claim that is being supported if the null hypothesis is rejected.

What is a type I error?

A type I error is the error of rejecting the null hypothesis when it is true.

What is a type II error?

A type II error is the error of failing to reject the null hypothesis when it is false.

Sure, here are some MCQs without mentioning the topic of variance:

The standard deviation is a measure of how spread out a set of numbers is. It is calculated by taking the square root of the variance.
The variance is a measure of how spread out a set of numbers is. It is calculated by taking the average of the squared deviations from the mean.
The mean is the average of a set of numbers. It is calculated by adding all the numbers in the set and dividing by the number of numbers in the set.
The median is the middle number in a set of numbers. If there are an even number of numbers in the set, the median is the average of the two middle numbers.
The mode is the most common number in a set of numbers.
The range is the difference between the largest and smallest numbers in a set of numbers.
The interquartile range is the difference between the upper and lower quartiles. The upper quartile is the median of the upper half of the numbers in a set, and the lower quartile is the median of the lower half of the numbers in a set.
The coefficient of variation is a measure of how spread out a set of numbers is relative to its mean. It is calculated by dividing the standard deviation by the mean.
The skewness of a set of numbers is a measure of how asymmetrical the distribution of the numbers is. A set of numbers with a positive skewness has a longer tail on the right, and a set of numbers with a negative skewness has a longer tail on the left.
The kurtosis of a set of numbers is a measure of how peaked the distribution of the numbers is. A set of numbers with a high kurtosis is more peaked than a set of numbers with a low kurtosis.

I hope this helps!