Statistics

Art of learning from data. Our primary intention is to find the tendency of the graph.

Types

1. Descriptive
2. Inferential

Descriptive Statistics

It provides summary statistics of data. Helps to quantitatively interpret the features of data. The measures for descriptive statistics are :

Measures of Central Tendency

Focus on the average or central point of a dataset.

• Mean
• Median
• Mode

Measures of Spread

Focus on the dispersion of data from the central point.

• Range
• Standard Deviation
• Variance
• Interquartile Range

Inferential Statistics

Makes inferences about the properties of a population. Makes propositions about a population. Some popular terms dealt with here are :

We make use of the population, statistics, sample and parameter.

Estimation

Process of analyzing the parameter of population.

Point Estimation

Single value that determines the parameter of population.

Confidence Intervals

Range of values within which the parameter is included.

Hypothesis Tests

Specific values of the parameter are tested.

Measures

Estimation

In this process, a sample is drawn from a population to estimate the following:

Parameter: Mean, Standard Deviation, Proportion, Correlation.

Confidence Interval: Range of values of an unknown population parameter.

Hypothesis Testing

A hypothesis test helps in determining whether to reject or retain a claim about a population, depending on the evidence provided by a sample of data.

Some basic Statistical concepts

Mean

The average over a set of values over a random variable x.

Median

The value occurring at the center of a sorted set of data describing the random variable x.

Mode

The value that appears the most in the set of data describing the random variable x.

Standard deviation

A measure of the degree of spread around the mean of a dataset. This is useful for explaining the quality of the spread of a dataset.

sigma(x) = sqrt( (sum(xi - mean(x))^2 ) / N )

This like covariance is dependent on an 'expectation', for which we most often make use of mean to make it reasonably easy to explain, but it just as easily could be anything else.

Co-Variance

Square of the deviation.

Advanced reading

Since statistics is to study the tendency of a graph, we actually define the tendency in terms of 'moment'. The concepts related to it are given here.

Moment

Quantitative measure related to the shape of the graph of a distribution(or a function). If the function represents mass then the first moment is the center of the mass, the second moment is the rotational inertia. If it is probability distribution, then the first moment is the expected value, the second central moment is variance, the third standardized moment is the skewness, the fourth standardized moment is the kurtosis.

Expected value

It is the generalization of the weighted average, and is intuitively the arithmetic mean for a large number of independent realizations of X. This is termed as the expectation, mean, average or first statistical moment.

Variance

It is the expected value of the squared deviation from the mean of X. i.e. Var(x) = E[(x - mean(x))^2] = sum((x - mean(x))^2) / n This can also be written as : Var(x) = sum(sum((xi - xj)^2) / n^2 Which means that is the squared deviations of all points from each other.

This is different for every kind of distribution. The following link gives a degree of verbosity to it.

Variance

The usual one we study are the formulae for normal distributions

Skewness

It is the measure of asymmetry of the probability distribution. It can be positive, negative, zero or undefined. A negative skew means the tail is more towards the left of the distribution, or the peak is shifted towards the positive x axis. A positive skew is vice versa.

Make use of Pearson's Moment Coefficient of Skewness.

Kurtosis

It is a measure of the shape of the tailedness,