Descriptive characteristics

The basic idea of statistics is simple: you want to extrapolate from the data you have collected to make general conclusions. Population can be e.g. all the voters and sample the voters you polled. Population is characterized by parameters and sample is characterized by statistics. For each parameter we can find appropriate statistics. This is called estimation. Parameters are always fixed, statistics vary from sample to sample.

Confidence interval

A 95% CI is the interval in which you are 95% certain contains the true population value. Another definition: Statistic calculated from 100 random samples will fall within a 95% CI in 95 times and be outside 5 times.

Central Tendency

Mean (Arithmetic mean)
This is the average of the data values:

Sample mean:

Geometric Mean

The geometric mean is a useful measure of central tendency for samples that are log-normally distributed (i.e. the logarithms of the observations are from an approximately normal distribution). The geometric mean can be calculated only from positive values.

Harmonic Mean

The harmonic mean is used to average rates. For example, suppose we want the average speed of a bus that travels a fixed distance every day at speeds s₁, s₂, and s₃. The average speed, found by dividing the total distance by the total time, is equal to the harmonic mean of the three speeds. The harmonic mean is appropriate when the distance is constant from trial to trial and the time required was variable. However, if the times were constant and the distances were variable, the arithmetic mean would have been appropriate.

Median

The median is the 50th percentile of the data set. It is the point that splits the database in half. Unlike mean, median is not affected by extreme values.

Mode

This is the most frequently occurring value in a data.

Variability (Dispersion)

Range
The difference between the largest and smallest values for a variable.

Interquartile Range

It is the difference between the third quartile and the first quartile (between the 75th percentile and the 25th percentile). This represents the range of the middle 50 percent of the distribution.

Variance

It is an average of the squared deviations from the mean.

Sample variance:

Standard Deviation

It measures the average distance between a single observation and its mean.

Standard Error of Mean

Coefficient of Variation

The coefficient of variation is a relative measure of dispersion. It is most often used to compare the amount of variation in two samples.

Shape

Skewness

This statistic measures the direction and degree of asymmetry. A value of zero indicates a symmetrical distribution. A positive value indicates skewness (longtailedness) to the right while a negative value indicates skewness to the left. In a perfectly symmetrical, non-skewed, distribution the mean, median and mode are equal. positive skewness indicates that mean is greater than median, which means that most of values are less than the mean. Negative skewness indicates that median is greater than mean, which means that most of values is greater than the average. Histograms (Figure 1-3) present triangular distributions with the same mean (10), variance (2) and kurtosis (2.4) with different skewness.

Figure 1 Symmetrical distribution

Figure 2 Distribution with positive skewness (0.55)

Figure 3 Distribution with negative skewness (–0.55)

There are two most frequently used measures of skewness. Pearson's b₁ and Fisher's g₁ are computed as follows:

where m_r are central moments:

Kurtosis

There are two most frequently used measures of kurtosis. Pearson's b₂ and Fisher's g₂ are computed as follows:

This statistic measures the heaviness of the tails of a distribution. The usual reference point in kurtosis is the normal distribution. If the b₂ equals 3 (g₂ zero) and the skewness is zero, the distribution is normal. Unimodal distributions that have kurtosis greater than 3 have heavier tails than the normal. These same distributions also tend to have higher peaks in the center of the distribution. Unimodal distributions whose tails are lighter than the normal distribution tend to have a kurtosis that is less than 3. In this case, the peak of the distribution tends to be broader than the normal. Histograms (Figure 4-7) present four symmetrical distributions with the same mean (0) and variance (5/3) with different kurtosis.

Figure 4 Normal distribution (b₂=3)

Figure 5 Student's t distribution with 5 degrees of freedom (b₂=9)

Figure 6 Triangular distribution (b₂=2.4)

Figure 7 Uniform distribution (b₂=1.8)

The measures of shape require more data to be accurate. For example, a reasonable estimate of the mean may require only ten observations in a random sample. The standard deviation will require at least thirty. A reasonably detailed estimate of the shape (especially if the tails are important) will require several hundred observations.