Statistics

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Statistics is a way to collect and analyze measurements. We use statistics to describe data and to test theories about the world and how it works. Statistics is based on probability — the "laws of chance".

Statistics can be divided into 3 portions:

1. Probability Distributions which are extended from Probability Theory from Mathematics
2. Descriptive Statistics - to describe the data collected through observations or experiments
3. Inferential Statistics - assume that the collected data are from a certain probability distribution, and based on that probability distribution attributes and properties, we can make the statistical inferences, such as estimation, prediction and forecasting

[edit] Collecting data

We have to find numbers (collect data) about the world before we can make statistics to describe it. Usually we want to study a population or a process. For example, we might want to know how many people in our country watch our favorite TV program, or whether a new drug will cure a disease.

We often gather data by doing a survey or an experiment. When we do a survey, we choose a small number of individuals from a population and collect data from them. For example, we might ask questions if they are people, or we might count or measure them if they are objects or animals. In an experiment, we want to know about the effect of some treatment. We now choose two small groups of individuals, and apply the treatment to one group before collecting data on the outcome. The other group is not treated (or is only given a fake treatment, a placebo), so we can tell how much effect the treatment had by comparing their outcomes with the treated group.

We hope the data we get from our sample is similar to the data from the whole population, but there are two kinds of problems we might have.

1. If the sample is very small, it is more likely that the sample will be very different from the population. This kind of error is called a chance error.
2. If we do not choose individuals randomly we might not be fair in how we choose the individuals, so the sample might be different from the population even if it was very big. This kind of error is called bias.

We can avoid chance errors by taking a larger sample, and we can avoid some bias by choosing randomly. However, sometimes large random samples are hard to take. And bias can happen if some people refuse to answer our questions, or if they know they are getting a fake treatment. These problems can be hard to fix.

[edit] Descriptive statistics

[edit] Finding the middle of the data

The middle of the data is often called an average. The average tells us about a typical individual in the population. There are three kinds of average that are often used: the mean, the median and the mode.

The examples below use this sample data:

 Name | A   B   C   D   E   F   G   H   I   J
---------------------------------------------
 score| 23  26  49  49  57  64  66  78  82  92

The formula for the mean is

Where are the data and N is the population size. (see Sigma Notation).

In our example

The median is the middle item of the data. To find the median we sort the data from the smallest number to the largest number and then choose the number in the middle. If there are an even number of data we choose the two middle ones and calculate their mean. In our example there are 10 data, the two middle ones are "E" and "F", so the median is (57+64)/2 = 60.5.

The mode is the most frequent item of data. For example the most common letter in English is the letter "e". We would say that "e" is the mode of the letters.

[edit] Finding the spread of the data

[edit] Other descriptive statistics

We use it to find out that some percent, percentile, number, or fraction of people or things in a group do something or fit in a certain category.

For example, social scientists used statistics to find out that 49% of people in the world are males.

See also: Normal distribution