# SECTION VI - CHAPTER 39 - Descriptive Statistics basics

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### SECTION VI - CHAPTER 39 - Descriptive Statistics basics

• 1. Module - 58 Descriptive Statistics Basics CMT LEVEL - I
• 2. Descriptive statistics • Descriptive statistics are brief descriptive coefficients that summarize a given data set, which can be either a representation of the entire or a sample of a population. • Descriptive statistics are broken down into measures of central tendency and measures of variability (spread). • Measures of central tendency include the mean, median, and mode • while measures of variability include the standard deviation, variance, the minimum and maximum variables, and the kurtosis and skewness.
• 3. Measures of Central Tendency Mean Median Mode Geometric Mean Weighted Arithmetic Mean
• 4. Mean (or average) •The mean (or average) is the most popular and well known measure of central tendency. It can be used with both discrete and continuous data, although its use is most often with continuous data • For example, consider the wages of staff at a factory below: •The mean salary for these ten staff is \$30.7k.
• 5. Median • The median is the middle score for a set of data that has been arranged in order of magnitude. The median is less affected by outliers and skewed data. In order to calculate the median, suppose we have the data below: •Our median mark is the middle mark - in this case, 56
• 6. Mode • The mode is the most frequent score in our data set. On a histogram it represents the highest bar in a bar chart or histogram. You can, therefore, sometimes consider the mode as being the most popular option. An example of a mode is presented below:
• 7. Geometric Mean • The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. • It is technically defined as "the nth root product of n numbers.“ • The geometric mean is an important tool for calculating portfolio performance for many reasons, but one of the most significant is it takes into account the effects of compounding. • For volatile numbers, the geometric average provides a far more accurate measurement of the true return by taking into account year-over-year compounding that smooths the average.
• 8. Weighted Mean •The weighted mean is a type of mean that is calculated by multiplying the weight (or probability) associated with a particular event or outcome with its associated quantitative outcome and then summing all the products together. • (5* 282.38+4*280.86+3*281.33+2*279.95+5*281.42)/ (5+4+3+2+1) • 4220.65/15 = 281.38 Days 1 2 3 4 5 Price 282.38 280.86 281.33 279.95 281.42
• 9. Measurement of Dispersion •the measure of dispersion shows the scatterings of the data. It tells the variation of the data from one another and gives a clear idea about the distribution of the data. •The measure of dispersion shows the homogeneity or the heterogeneity of the distribution of the observations. •A measure of dispersion should be rigidly defined •It must be easy to calculate and understand •Not affected much by the fluctuations of observations •Based on all observations
• 11. Standard deviations • A standard deviation is the positive square root of the arithmetic mean of the squares of the deviations of the given values from their arithmetic mean. • It is denoted by a Greek letter sigma, σ. • The square of the standard deviation is the variance. It is also a measure of dispersion. • Squaring the deviations overcomes the drawback of ignoring signs in mean deviations • Suitable for further mathematical treatment • Least affected by the fluctuation of the observations • The standard deviation is zero if all the observations are constant • Independent of change of origin
• 12. Variance • Variance (σ2) is a measurement of the spread between numbers in a data set. • It measures how far each number in the set is from the mean and is calculated by taking the differences between each number in the set and the mean, squaring the differences (to make them positive) and dividing the sum of the squares by the number of values in the set. • Variance is one of the key parameters in asset allocation. • The variance of asset returns helps investors to develop optimal portfolios by optimizing the return-volatility trade- off in investment portfolios.
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