Statistical Concepts And Market Returns

## Quantitative Methods

• #### Introduction

Statistics is the study of the collection, organization, analysis, interpretation and presentation of data. It deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments. When analyzing data, it is possible to use one of two statistics methodologies: descriptive statistics or inferential statistics.

#### Descriptive statistics

Descriptive statistics is a collection of information, or the quantitative description itself. It is distinguished from inferential statistics in that descriptive statistics aim to summarize a sample. Even when a data analysis draws its main conclusions using inferential statistics, descriptive statistics are generally also presented. For example in a newspaper reporting on a study involving human subjects, there typically appears a table giving the overall sample size, sample sizes in important subgroups and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, and the proportion of subjects with related comorbidities.

Some measures that are commonly used to describe a data set are measures of central tendency and measures of variability or dispersion. Measures of central tendency include the mean, median and mode, while measures of variability include the standard deviation (or variance), the minimum and maximum values of the variables, kurtosis and skewness.

#### Inferential Statistics

Inferential statistics are tools used to draw larger generalizations from observing a smaller portion of data. In basic terms, descriptive statistics intend to describe. Inferential statistics intend to draw inferences, the process of inferring.

## Data collection

#### Sampling

In case census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Statistics provides tools for prediction and forecasting the use of data through statistical models.Representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually representative. Statistics offers methods to estimate and correct for any random trending within the sample and data collection procedures.

Sampling theory is part of the mathematical discipline of probability theory. Probability is used in "mathematical statistics" to study the sampling distributions of sample statistics and, more generally, the properties of statistical procedures. The difference in point of view between classic probability theory and sampling theory is, roughly, that probability theory starts from the given parameters of a total population to deduce probabilities that pertain to samples. Statistical inference, however, moves in the opposite direction—inductively inferring from samples to the parameters of a larger or total population.

#### Experimental and observational studies

Objective of a statistical research project is to investigate causality, and  to draw a conclusion on the effect of changes in the values of predictors . There are two major types of causal statistical studies: experimental studies and observational studies. In both types of studies, the effect of differences of an independent variable  on the behavior of the dependent variable are observed. The difference between the two types lies in how the study is actually conducted. Each can be very effective. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In an observational study data are gathered and correlations between predictors and response are investigated. While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data – like natural experiments and observational studies – for which a statistician would use a modified, more structured estimation method (e.g., Difference in differences estimation and instrumental variables, among many others) that will produce consistent estimators

#### Scales of Measurement in Statistics

Measurement scales are used to categorize and/or quantify variables. This lesson describes the four scales of measurement that are commonly used in statistical analysis: nominal, ordinal, interval, and ratio scales.

#### Properties of Measurement Scales

Each scale of measurement satisfies one or more of the following properties of measurement.

*Identity: Each value on the measurement scale has a unique meaning.
*Magnitude: Values on the measurement scale have an ordered relationship to one another. That is, some values are larger and some are smaller.
*Equal intervals. Scale units along the scale are equal to one another. This means, for example, that the difference between 1 and 2 would be equal to the difference between 19 and 20.
*A minimum value of zero. The scale has a true zero point, below which no values exist.

#### Nominal Scale of Measurement

The nominal scale of measurement only satisfies the identity property of measurement. Values assigned to variables represent a descriptive category, but have no inherent numerical value with respect to magnitude.

Gender is an example of a variable that is measured on a nominal scale. Individuals may be classified as "male" or "female", but neither value represents more or less "gender" than the other. Religion and political affiliation are other examples of variables that are normally measured on a nominal scale.

#### Ordinal Scale of Measurement

The ordinal scale has the property of both identity and magnitude. Each value on the ordinal scale has a unique meaning, and it has an ordered relationship to every other value on the scale.

An example of an ordinal scale in action would be the results of a car  race, reported as "win", "place", and "show". We know the rank order in which cars finished the race. The car that won finished ahead of the car that placed, and the car that placed finished ahead of the car that showed. However, we cannot tell from this ordinal scale whether it was a close race or whether the winning car won by a mile.

#### Interval Scale of Measurement

The interval scale of measurement has the properties of identity, magnitude, and equal intervals.

A perfect example of an interval scale is the Fahrenheit scale to measure temperature. The scale is made up of equal temperature units, so that the difference between 40 and 50 degrees Fahrenheit is equal to the difference between 50 and 60 degrees Fahrenheit.

With an interval scale, you know not only whether different values are bigger or smaller, you also know how much bigger or smaller they are. For example, suppose it is 60 degrees Fahrenheit on Monday and 70 degrees on Tuesday. You know not only that it was hotter on Tuesday, you also know that it was 10 degrees hotter.

#### Ratio Scale of Measurement

The ratio scale of measurement satisfies all four of the properties of measurement: identity, magnitude, equal intervals, and a minimum value of zero.

The weight of an object would be an example of a ratio scale. Each value on the weight scale has a unique meaning, weights can be rank ordered, units along the weight scale are equal to one another, and the scale has a minimum value of zero.

Weight scales have a minimum value of zero because objects at rest can be weightless, but they cannot have negative weight.