Statistics ch3 -Summary measures

Arithmetic mean

is the primary measure of central location

• Referred to as the mean or the average

Median

is a measure useful when outliers are present.

The median is the middle observation in a data set.

• It divides the data in half: an equal number of observations lie above and below the median.

One mode

Unimos al

Mode

is the only meaningful measure of central location to summarize a categorical variable

skewness coefficient

is a numerical measure of skewness.

• Skewness = 0: Observations are evenly distributed on both sides of the mean.

• Skewness > 0: Extreme observations are concentrated in the right tail of the distribution.

• Skewness < 0: Extreme observations are concentrated in the left tail of the distribution

weighted mean

is relevant when some observations contribute more than others.

Pth percentile

divides the data into two parts.

• Approximately p percent of the observations are less than the pth percentile

five-number summary

for a variable is commonly reported.

• It includes the minimum value, quartiles, and the maximum value.

Range

the difference between the maximum and minimum value.

• Range = Max − Min

Interquartile

is the difference between the third quartile and the first quartile.

• IQR = Q3 – Q1

Mean absolute difference

is the average of the absolute differences between the observations and the mean

Variance

is defined as the average of the squared differences between the observations and the mean

Standard deviation

is the positive square root of the variance.

• Provides a remedy to the issues related to variance.

Coefficient Variation

is a relative measure of dispersion.

Mean-variance analysis

postulates that the performance of an asset is measured by its rate of return evaluated in terms of reward (mean) and risk (variance)

Sharpe ratio

is the “reward-to-variability” ratio.

Sharpe ratio characterizes how well the return compensates for the risk and measures the extra reward per unit of risk.

Chebyshev’s Theorem:

The proportion of observations that lie within k standard deviations from the mean is at least 1 - 1/k2 (k is any number greater than 1).

Empirical rule

makes more precise statements about
the percentage of observations that fall within certain
intervals for symmetric and bell-shaped distributions:
• Shape can be detected using a histogram,

Z-score

is a way to graphically display a five-number
summary.

Boxplot

box-plot positively skewed

median is left of center and the right
whisker is longer than the left whisker

box-plot negatively skewed

median is right of center and the left
whisker is longer than the right whisker

box-plot symmetry

median is in the center of the box, and the left
and right whiskers are equally distant from their respective quartiles

correlation coefficient

describes both the direction and the strength of the linear relationship between x and y.