CFA Exam: Dec. 2015 Level I > Study Session 2. Quantitative Methods: Basic Concepts > Reading 7. Statistical Concepts and Market Returns > LOS 7.e.m.

- N = the number of observations in the entire population.
- X
_{i}= the ith observation - ΣX
_{i}= add up X_{i}, where i is from 0 to N.

- If the data set encompasses an entire population, the arithmetic mean is called a population mean.
- If the data set includes a sample of values taken from a population, the arithmetic mean is called a sample mean.

It is the most widely used measure of central tendency. When the word "mean" is used without a modifier, it can be assumed that it refers to the arithmetic mean. The mean is the sum of all the scores divided by the number of scores. It is used to measure prospective (expected future) performance (return) of an investment over a number of periods.

- All interval and ratio data sets (e.g. incomes, ages, rates of return) have an arithmetic mean.
- All data values are considered and included in the arithmetic mean computation.
- A data set has only one arithmetic mean. This says that the mean is unique.
- The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero.
**Deviation**from the arithmetic mean is the distance between the mean and an observation in the data set.

The arithmetic mean has the following disadvantages:

- The mean can be affected by extremes, that is, unusually large or small values.
- The mean cannot be determined for an open-ended data set (i.e., n is unknown).

- It exists only if all the observations are greater than or equal to zero. In another word, it cannot be determined if any value of the data set is zero or negative.
- If values in the data set are all equal, both the arithmetic and geometric means will be equal to that value.
- It is always less than the arithmetic mean if values in the data set are not equal.

It is typically used when calculating returns over multiple periods. It is a better measure of the compound growth rate of an investment. When returns are variable by period, the geometric mean will always be less than the arithmetic mean. The more disperse the rates of returns, the greater the difference between the two. It is not so highly influenced by extreme values as is the arithmetic mean.

The mean, median, and mode are equal in symmetric distributions. The mean is higher than the median in positively skewed distributions and lower than the median in negatively skewed distributions. Extreme values affect value of mean, while median is less affected by outliers. Mode helps to identify shape and skewness of distribution.

MURF: how do you do a geometric mean with a negative number?? 5, -3, 6 |

chandos: it cannot be calculated if one of the values is -ve |

aggieguy05: it can too be done. (1.05*.97*1.06)=g^3 |

db28luke: you add 1 to each return and take the nth root minus 1 |

db28luke: its in the book...page 125 |

mogulcn: in the case above, it is still positive as the data set are 1+Rt. in the book, it is said that the observations will never be negative becasue the biggest negative return is -100% |

achu: think of geometric mean as something like "multiplicative mean" average- product of n items then taken to 1/n th power. |

valerycfa: When calculating variance, why do we loose a degree of freedom when passing from population to sample calculation ? |

Mariecfa: If the sample variance were defined with division by n, it would systematically underestimate the value of the population variance. So, we compensate by increasing its overall value by making its denominator smaller (by using n-1 instead of n). Division by (n-1) causes the sample variance to target the value of the population variance, whereas division by (n) causes the sample variance to underestimate the value of the population variance. |

AmyJ: How do you solve for Geometric mean with an HP 12C calculator? Thank you. |

boddunah: amyJhp 12c platinum solution for geometric return. for example yearly returns are 5%,(3%),2% geometric return as follows step 1 :1.05*0.97*1.02 = 1.038870. (3% is negative return. so 1-0.03=.97) step 2: enter 3 and press button 1/x .result = 0.3333.(we used 3 bc 3 years returns were given) step 3 :0.3333 (already entered) press button Y^x. It should give you 1.0128. step 4 : subtract 1 from 1.0128 = 0.0128 0r 1.28% geometric return. |

jpducros: What do we use an harmonic mean for ? |

knowles242: as jpducros indicated is there an application of the harmonic mean? |

Mosobalaje: Harmonic mean is generally used to measure average investment costs over a time period. It's not used to calculate returns. |

Barchie: Why is it that the geometric mean is not as affected by the extremes? (that is it's advantage, I just don't get why not.) |

birdperson: 2 other "fun" facts-- sum of deviations from the (arithmetic) mean = 0 -- when the values are positive and not equal, H < G < A |