• Mean: The "average" number; found by adding all data points and dividing by the number of data points.

• Median: The middle number; found by ordering all data points and picking out the one in the middle (or if there are two middle numbers, taking the mean of those two numbers).

• Mode: The most frequent number—that is, the number that occurs the highest number of times.


mean, median and mode

• Variance: the average squared distance between the mean and each data value, $$\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2$$, where $$\mu$$ is the mean and $$N$$ is the number of data values. Because the deviations are squared, variance is expressed in squared units. • Standard deviation: the square root of the variance, $$\sigma = \sqrt{\sigma^2}$$, which returns the measure of spread to the same units as the original data.

variance and standard deviation

In statistics, consistency is a property of an estimator where its probabilistic limit converges to the true underlying parameter value as the sample size approaches infinity ($$	ext{plim}_{mightarrow\infty}\hat{	heta}_m=	heta$$). When an estimator is consistent, accumulating more data guarantees that empirical estimates will eventually match the true theoretical probabilities.

Consistency of an Estimator

The Mean Squared Error (MSE) of an estimator $$\hat{\theta}_m$$ with respect to a parameter $$\theta$$ is the expected value of the squared error:

$$MSE(\hat{\theta}_m) = \mathbb{E}((\hat{\theta}_m - \theta)^2)$$

It can also be expressed as the sum of the squared bias of the estimator and its variance:

$$MSE(\hat{\theta}_m) = Bias^2(\hat{\theta}_m) + Var(\hat{\theta}_m)$$

Mean Squared Error

• mean

• median

• mode

• percentiles

• range

• variance

• standard deviation

• mean squared error

• consistency

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University of California, Davis

- Number of data points (instances)
- Attributes (variables)
- Types of attributes (variables)
- Number of attributes (variables)
- Statistical measures

Characteristics of a dataset

Most statistics and data science textbooks use the letter $$n$$ to represent the number of data points (instances), and sometimes, they call it the number of observations. The number of distinct data points depends on how the unit of data is defined in the dataset. In tabular datasets, this number corresponds to the number of rows.

Number of data points (instances)

A datapoint, instance, sample, or observation, is a unit of data in a dataset. When datasets are tabular, $$x_{ij}$$ usually represents the value of the $$j$$th variable (attribute) in the $$i$$th data point, where $$i=1,2,...,n$$ indicates the index of the data point and $$j=1,2,...,p$$ identifies the variable. This way, a dataset can be represented as an $$n\times p$$ matrix $$X$$, where the row (or vector) $$x_i$$ represents a datapoint that consists of the $$p$$ values of $$x_{i1}, x_{i2}, x_{i3}, \dots , x_{ip}$$ corresponding to each variable. $$X=\begin{bmatrix} x_{11} & x_{12} & x_{13} & \dots & x_{1p} \\ x_{21} & x_{22} & x_{23} & \dots & x_{2p} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{n1} & x_{n2} & x_{n3} & \dots & x_{np} \end{bmatrix} =\begin{bmatrix} x_1^T \\ x_2^T \\ \vdots \\ x_n^T \\ \end{bmatrix}$$

Datapoint

Most statistics and data science textbooks use the letter p to represent this number, and sometimes, they call it the number of covariates.
In tabular datasets, this number is the number of columns.

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