The basics

Credits & Sources:

• What would be some examples of when the "mean" would be preferred over the "median"?

• Mean, Median, Mode: What They Are, How to Find Them (Statistics How To)

• Mean (Wikipedia)

• Median (Wikipedia)

• Range (Wikipedia)

• Mode (Wikipedia)

• Variance (Wikipedia)

• Standard deviation (Wikipedia)

• Standard eror (Wikipedia)

• Statistical mean, median, mode and range (Margaret Rouse, Tech Target)

• Mean vs. Median: When to Use? (Stack Exchange)

• Standard deviation and Variance (Math is Fun)

• Understanding Principal Components Analysis (Rishav Kumar)

Properties of distributions

Mean

Discrete distribution

Found by adding all of the numbers together and dividing by the number of items in the set:

$${\bar {x}}={\frac {1}{n}}\left(\sum _{i=1}^{n}{x_{i}}\right)={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}$$

Continuous distribution

It's obtained by integrating the product of the variable with its probability as defined by the distribution. Given the $f(x)$ continuous probability distribution:

$$\mu = \int _{-\infty }^{\infty }xf(x)\,dx$$

Types

• Geometric mean
• Harmonic mean
• ...

Median

Discrete distribution

It depends on whether the number of terms in the distribution. Once the values are sorted.

• If the given number of terms is odd:

  • It's the value in de middle.

  • 1, 3, 3, 6, 7, 8, 9 $\Rightarrow$ 6

• If the given number of terms is even:

  • It's the average of the two terms in the middle.

  • 1, 2, 3, 4, 5, 6, 8, 9 $\Rightarrow$(4+5) $\div$ 2 = 4.5

Continuos distribution

It's the value m such that the probability is at least 0.5 that a randomly chosen point on the function will be less than or equal to m.

$$\operatorname {P} (X\geq m)=\operatorname {P} (X\leq m)=\int _{-\infty }^{m}f(x)\,dx={\frac {1}{2}}$$

Mode

Discrete distribution

It's the most repeated value within the distribution. Example:

• 1, 2, 2, 3, 4, 7, 9 $\Rightarrow$ 2

It's quite common to find more than one mode, especially if there aren't many terms. A distribution with two modes is called bimodal. As well, with three it's called trimodal.

Continuous distribution

It's the maximum value of the function. As with discrete distributions, there may be more than one mode.

Range

Discrete distribution

It's the difference between the maximum value and the minimum value. Example:

• 1, 2, 2, 3, 4, 7, 9 $\Rightarrow$ 9 - 1 = 8

Continuous distribution

It's the difference between the two extreme points on the distribution curve. For any value outside the range of a distribution, the value of the function is equal to 0.

Source: Siyavula

Variance

It measures how far a set of random numbers are spread out from their mean.

Population Variance

$$\sigma^2 = \frac {1}{N} \sum _{i=1}^{N} (x_{i} - \mu)^2$$

Sample Variance

When data is a sample, this formula is used as a "correction" over the original one.

$$s^2 = \frac {1}{N-1}\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2}$$

Example Numpy

import numpy as np

np.random.seed(333)
np.std(np.random.rand(100))

Standard deviation

It's is the squarred root of the variance. A low standard deviation indicates that the data points tend to be close to the mean.

Population Std

$$\sigma = \sqrt{ \frac {1}{N} \sum _{i=1}^{N} (x_{i} - \mu)^2}$$

Sample Std

When data is a sample, this formula is used as a "correction" over the original one:

$$s = {\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2}}}$$

Example with Numpy

import numpy as np

np.random.seed(333)
np.var(np.random.rand(100))

Covariance

It's a measure of the joint variability of two random variables. measures of the extent to which corresponding elements from two sets of ordered data move in the same direction. It measures how much two variables vary together. It’s similar to variance, but where variance tells you how a single variable varies, covariance tells you how two variables vary together.

Population Covariance

$$\sigma_{XY} = \frac{ \sum_{(x,y) \epsilon S} (x - \mu_X) (y - \mu_Y) }{N}$$

Sample Covariance

$$cov(x, y) = \frac{ \sum_{i=1}^N (x_i - \overline{x}) (y_i - \overline{y}) }{N - 1}$$

Covariance vs Correlation Matrix

Related questions

Mean vs Median

• Median is much less sensitive to outliers.

• However, almost all analytic calculations on sets of data are more natural in terms of the mean than the median.

• The difference between the median and the mean is useful to represent how skewed the data is.

• The real use of the median comes when the data set may contain extreme outliers. Then, describing the distribution in terms of quartiles can be more informative than quoting $\mu$ and $\sigma$.

• For skewed distributions, the mean is not necessarily the same as the median or the mode. For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. Median and mode are often more intuitive measures for such skewed data, BUT many skewed distributions are in fact best described by their mean, including the Exponential and Poisson distributions.

Comparison of mean, median and mode of two log-normal distributions with different skewness.

Difference between standard deviation of a sample and standard error of the population mean

Meanwhile the standard deviation expresses how disperse is data with respect to the mean, the standard error measures the standard deviation of its sampling distribution.

The sampling distribution of a population mean is generated by repeated sampling and recording of the means obtained. This forms a distribution of different means, and this distribution has its own mean and variance.

Standard Error of a Population

Given the standard error of the population $\sigma$ and the size $n$ of a sample, the standard error of a sample of this population is expressed as:

$$\sigma_{x}^{-} = \frac{\sigma}{\sqrt{n}}$$

Standard Error of a Sample

Since the population standard deviation is seldom known, the standard deviation of a sample is used to approximate this statistic:

$$s_{x}^{-} = \frac{s}{\sqrt{n}}$$