Question

1.What is the central limit theorem and why is it important in statistics? 2.Explain the differences between the mean, mode and median. Which is the most useful measure of an average and why? 3.Which is a more useful measure of central tendency for stock returns – the arithmetic mean or the geometric mean? Explain your answer.

Answer 1

Central limit theorem

The central limit theorem states that when samples from a data set with a known variance are aggregated their mean roughly equals the population mean. In other words, CLT is a statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population.

Importance of Central limit theorem

Many practices in statistics, such as those involving hypothesis testing or confidence intervals, make some assumptions concerning the population that the data was obtained from. One assumption that is initially made in a statisticscourse is that the populations that we work with are normally distributed. The use of an appropriate sample size and the central limit theorem help us to get around the problem of data from populations that are not normal.Thus, even though we might not know the shape of the distribution where our data comes from, the central limit theorem says that we can treat the sampling distribution as if it were normal.

Difference between mean , median and mode

The mean is the average of all numbers and is sometimes called the arithmetic mean. To calculate mean, add together all of the numbers in a set and then divide the sum by the total count of numbers.

The statistical median is the middle number in a sequence of numbers. To find the median, organize each number in order by size; the number in the middle is the median.

The mode is the number that occurs most often within a set of numbers.

Difference between them can be understood as :

There can often be a "best" measure of central tendency with regards to the data you are analysing, but there is no one "best" measure of central tendency. This is because whether you use the median, mean or mode will depend on the type of data you have, such as nominal or continuous data; whether your data has outliers and/or is skewed; and what you are trying to show from your data.

The mean is usually the best measure of central tendency to use when your data distribution is continuous and symmetrical, such as when your data is normally distributed.

The mode is the least used of the measures of central tendency and can only be used when dealing with nominal data.

The median is usually preferred to other measures of central tendency when your data set is skewed.

More useful measure of central tendency for stock returns

If any one of the scores is zero then the geometric mean is zero. The geometric mean does not make sense if any scores are less than zero. In that case, more useful measure of central tendency for stock returns is arithmetic mean. if the observations are non-zero then one can use geometric mean.The geometric mean is less affected by extreme values than is the arithmetic mean and is useful as a measure of central tendency for some positivelyskewed distributions.