# Statistics

February 27, 2019

Tags:

A link to notes (PDF) I had when I took a statistics unit a while ago. Not guaranteed to be correct.

# Introduction

Statistics the science of generalizing knowledge from data. Population complete set of elements being studied.

Variations in sampling

• Sampling variation/Intrinsic variation :: is variation caused by sampling
• Variability :: vary from the mean

z critical confidence interval critical value

## Normal distribution

Why normal distribution is important:

• Empirical rule
• Underlies distribution of P values which are used to test hypothesis

# Measures of central tendency

## Mean

### Sample mean

A point estimator of the population mean:

Most repeated

## Median

Value in the middle

## Variance

Applications in:

• QA
• Ops

In finance risk is often another term for a stock's variance. Some stocks are steady (low risk) but offer lower potential returns. Others are swing wildly (high risk) but offer more potential upside.

## Standard deviation

Measures the average distance your data values are from the mean. It's also \sqrt{variance}.

Closely grouped data has a large standard deviation and the opposite for spread out data.

It is:

• never negative
• never 0 unless there's no deviation at all in the data
• Greatly affected by outliers

### Sample standard deviation

Lower case /s/ means sample standard deviation

In sample standard deviation we do /n-1/ to overestimate the variation because /-1/ decreases the denominator making the result, s, bigger.

Simpler formula:

or

### Population standard deviation

Symbol for population standard deviation is \sigma. \ \mu is population mean.

We don't divide by n-1 but N because we don't want to overestimate our population.

## Empirical rule

How much proportion or percentage of a dataset will fall within certain std devs from the mean. Applies only to a normally distributed dataset. Also called 68%, 95%, 99.7% rule. \ If data is normally distributed then:

• 68% of data will fall within 1 standard deviation from the mean
• 95% of data will fall within 2 standard deviation from the mean
• 99.7% of data will fall within 3 standard deviation from the mean

Data values /within/ 2 standard deviations are usual. Data values /outside/ 2 standard deviations are unusual. A data value outside of 3 standard deviations from the mean is extremely rare.

Given different standard deviations (with different units, values, samples etc), we have to find a way to represent what has more spread. To do this we use:

• Coefficient of variation :: translates s in comparison to \bar x as a percentage
• Z score :: The number of standard deviations away from the mean that a data value lies in. This lets to compare two datasets directly to see which has more variation.

## Sampling distribution of sample mean

When we take many samples of the same size from a population and find the sample means \bar x. The means of those samples follow a normal curve when placed in their own distribution.

## Sampling distribution of sample variance

When we take many samples of the same size from a normal population and then fine those sample variances s^2, those sample variances don't follow a normal curve when placed in their own distribution.

They follow the chi-square \chi^2 distribution with n-1 degrees of freedom.

## Chi Squared distribution

Compares sample variance to pop variance. We try to estimate population variance

• n :: sample size
• $s^2$ :: sample variance
• $\sigma^2$ :: population variance

A \chi^2 distribution has a tail to the right.

1. Not symmetrical
2. Values are non negative - No 0 in the middle because std and variance can't be 0
3. As degrees of freedom go up the distribution becomes more symmetrical
4. Gives critical values for the area to the right - Based on area to the right

### Anatomy of the chi-square distribution

1. There is no one chi distribution
2. Area (probability) under the curve is 1
3. The curve is asymptotic; never touches the x axis
4. 1 is at the left and 0 is at the right
5. Cumulative probability runs right to left
6. Probabilities are found in the chi-square table in the same manner as normal curves

### Example 1

Given n = 12 and confidence level = 95%. Find the critical value which make the $\chi^2$ distribution.

Solution

Because we try to estimate pop variance then

We have two $\chi^2$ values A $\chi^2_{left}$ and $\chi^2_{right}$.

From above it's $\chi^2_{left} = 3.816 \chi^2_{right} = 21.29$.

The $\chi^2_{right}$ is larger therefore when $(n-1)s^2$ is divided by it, we get a smaller value. Therefore:

Variance:

$\dfrac{(n-1)s^2}{\chi^2_{right}} < \sigma^2 < \dfrac{(n-1)s^2}{\chi^2_{left}}$

Standard deviation:

$\sqrt{\dfrac{(n-1)s^2}{\chi^2_{right}}} < \sigma < \sqrt{\dfrac{(n-1)s^2}{\chi^2_{left}}}$

The $\alpha$ and confidence level are complimentary

This says our pop variance lies within this range with 95% certainty

### Example 2

We sample 10 phone chargers and we have a std dev of 0.15 volts. Construct a 95% CI for $\sigma$ and $\sigma^2$.

Solution

For voltage specifically use the sqaure root to get: $0.1031 < \sigma < 0.2738$

## F ratio and F distribution

Whether 2 sample variances are equal given the limits of random sampling We want to know whether a difference is statistically significant or caused by a sampling error.

#### F distribution

The distribution of F ratios sample df = n - 1

#### Equality of variance

Are the variances equal or not?

# Measures of relative standing

Comparing measures between or within datasets. This lets you compare the variation of two samples or populations.

## Coefficient of variation

The ratio of standard deviation to the mean as a percentage

## Z score

The number of standard deviations that data value is away from the mean. Same for sample as well as population. Z scores can be negative or positive. A z score at the mean is 0 Z scores can also be usual >= -2 && <= 2 or unusual < -2 && > 2 The larger the z score in terms of absolute value the more rare the data.

Sample

Population

## Quartiles

Data has to be sorted, has to be values. Go from left to right:

• Q1 :: bottom 25%
• Q2 (median) :: bottom 50%
• Q3 :: bottom 75% of the data There's no Q4 because that's everything.

|---|---|---|----|----|----|----|----| | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | |---|---|---|----|----|----|----|----|

In the sample above:\ Q1 = 4.5\ Q2 = 12.5\ Q3 = 24.5

|---|---|---|----|----|----|----|----|----| | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 39 | |---|---|---|----|----|----|----|----|----|

We pretend in this case that the 15 doesn't exist.\ Q1 = 4.5\ Q2 = 15\ Q3 = 32

## Percentiles

Separates data into 100 parts. We have 99 parts

## InterQuartile range (IQR)

75th percentile - 25th percentile

# Population vs sample data

## Population data

The population is defined by the researcher e.g all women, all bulbs produced by a certain company etc. Populations can be large, it's hard to collect data on each member of a population.

Collecting data on all members of a population is called a [[https://en.wikipedia.org/wiki/Census][census]].

## Sample data

When we need to make a conclusion about our population we use a sample. A small but well chosen sample can accurately represent the population.

Sample guidelines:

• All elements in the sample should be part of the population
• The sample should be representative of the population
• In most cases (if not all) samples from the population should be independent of each other

Kinds of samples:

• Random

A sample is always an approximation of the population. Therefore:

• Sample data sets always have error built into them

## Point estimate

Using a single value from a sample to approximate an entire population parameter.

You have no idea how accurate the point estimate is.

## Confidence interval

Range of numbers used to estimate a population parameter. Estimates a population proportion from a sample proportion.

They have:

1. Confidence level (1-\alpha): how confident you are that the actual value of the population parameter will be inside the interval.
• \alpha is the complement of the confidence level.
• most common confidence levels:
• 90% \rightarrow \alpha = 0.1
• 95% (most used) \rightarrow \alpha = 0.05
• 99% \rightarrow \alpha = 0.01

Requirements:

1. A random sample.
2. Conditions for binomial
• Fixed number of trials
• Trials are independent
• Two outcomes (success or failure)
• np >= 5 and nq >= 5

Example: The 95% CI for p is 0.38 < p < 0.497 I don't know what p is but I'm 95% sure that it falls between 0.38 and 0.497 of the population

## Critical value

A z score that separates the likely region from the unlikely region.

## Margin of error

Max difference between ^{p} and p.

### Example

We are given that n = 670, ^{p} = 0.85, we also jusqt learned that the standard eror of the sample proportion is SE = p(1-p). Which of the below is the correct calculation of the 95% confidence interval?

# Descriptive statistics

## z-scores

Measure of distance from the mean. \ How far from the mean is a is a given data point. How many standard deviations away (above or below) from the mean is a data point. Standard deviation here is a unit of measurement like a kg, meter etc.

z-scores are standardized measures where the unit is a standard deviations.

z score of the mean is 0 because it's zero distance from itself.

Like mass of person x is 5 kgs; we can say z-score of x is 1 standard deviations.

# Correlation

Independent variable should be on the x axis while the dependent variable should be on the y axis. Correlation seeks a statistical relationship between two variables or bivariate data.

A regression model is unique to the data it represents. Adding data will change the regression model. It's not proper to extrapolate above or below data being evaluated. How much better is our line of fit compared to only using the mean of the dependent variable.

## Correlation and causation

It is tempting to assume that one variable causes another however, correlation doesn't imply causation.

## Uses of correlation

• comparing models :: You compare a given model against just the mean of the dependent variable
• prediction :: useful because they indicate a predictive relationship that can be exploited in practice

## Correlation coefficient (r)

Correlation coefficient is a popular way of summarizing a scatter plot into one value between 1 and -1.

• +ve slope is +ve correlation
• -ve slope is -ve correlation
• 0 slope is no correlation

/A weak correlation is closer to 0; whereas a strong correlation is near 1 or -1/

## Steps

### 1. Fit line

Helps fit a straight line through the data Minimum square distances between fitted line and individual points

### 2. Remember slope

Remembers if slope is pointing upwards or downwards

### 3. Quality of fit of the straight line for the data

Shows how well the slope fits the data based on whether the correlation is weak or strong

## Example

Trying to see whether more fertilizer leads to higher yields of beans

|------------------|---|---|---|---|---|---|---| | Fertilizer (lbs) | 2 | 1 | 3 | 2 | 4 | 5 | 3 | |------------------|---|---|---|---|---|---|---| | Bushels of beans | 4 | 3 | 4 | 3 | 6 | 5 | 5 | |------------------|---|---|---|---|---|---|---|

xyx \cdot yx^2y^2
248416
13319
3412916
23649
46241636
55252525
3515925

## Residuals/errors

These are values of how far our values are from the line of best fit

## Coefficient of determination

Calculated by: r^2 = SSR/SST

When r^2*100 we get the percentage of results due to SSE

# Simple linear regression

We compare a model of the dependent variable on it's own against a model of the dependent variable against the independent variable.

Assume:

• normal distribution
• both x and y are continuous
• both x and y are numerical

## Coming up with a regression line

Get the centroid point made by ( (\bar x, \bar y) ). Your line of best fit must pass through the centroid.

TODO

## SST (Total Sum of Squares)

$$\Sigma (x_i - \bar x)^2$$

## SSR (Sum of squares due to regression)

SSR = Sum of squared errors of \bar y alone - Sum of squared errors of best line of fit

# Logistic regression

Dependent variable is binary We want to link our probabilities back to 0 & 1

Logistic regression seeks to:

1. Model the probability of an event occurring based on the values of an independent variable, which can be categorical or numerical.
2. Estimate the prob that an event occurs for a randomly selected observation vs the prob that the event doesn't occur
3. Predict the effect of a series of vars on a binary response var
4. Classify observations by estimating the prob that an observation is in particular category (e.g bank loan approved or not)

## Odds

Odds are probability of something occurring / probability of something not occurring

$$odds = \dfrac{P(occurring)}{P(not\ occurring)}$$

Probability of it not occurring is: 1 - probability of it occurring

$$odds = \dfrac{p}{q} = \dfrac{p}{1-p}$$

/What about events that have a probability of 1 occurring? We get odds of infinity/

### Examples

#### Flipping a fair coin

odds(heads) = \dfrac{0.5}{0.5} = 1 or 1:1

#### Rolling a fair die

Odds of getting 1 or 2:

odds(1 or 2) = \dfrac{0.333}{0.666} = \dfrac{1}{2} = 0.5 or 1:2

#### Deck of playing cards

Odds of pulling out a diamond card:

odds(diamonds) = \dfrac{0.25}{0.75} = \dfrac{1}{3} = 0.333 or 1:3

/There are 52 cards in a deck and 4 types of cards (diamond, spade, flowers & hearts) and in equal numbers/

## Odds ratio

A ratio of two odds We are comparing the likelihood of getting an outcome in two separate "systems"

If we want to know how much we increase the odds of getting an outcome by changing one variable and holding all others constant. /odds ratio for a variable show how the odds change with 1 unit increase in that variable holding all other variables constant./ e.g

• What are the odds of getting a loan approved by increasing your credit score by 1?
• What are the odds of getting a heart attack when you increase your bodyweight 1 kg past a certain threshold?

#### Examples

Say we want to start a casino and want to make some loaded coins to make sure the house wins. We may want to know how to load our coin so that the house wins but the players also win a few times to keep them coming. We want to know how many more times our loaded coin will get a certain outcome compares to a fair coin.

###### A fair coin

Odds ratio would be: \dfrac{2.333}{1.0} = 2.333

This means that in the loaded coin we are 2.333 more times likely to get heads than on the fair coin. Loading the coin by 2 increases the odds of getting a heads by 2.333

## Odds ratio in logistic regression

The odds ratio for a variable in logistic regression represents how the odds change with 1 unit increase in that variable holding all other variables constant.

### Example

By increasing our credit score by one how do we affect the probability of getting a loan approved?

Body weight and sleep apnea. Categories:

• apnea
• no apnea

Weight variable has an odds ratio of 1.07

This means a 1 pound increase in body weight increase the odds of having sleep apnea by 1.07.\ A 10 lbs increase in weight increase the odds to 1.98.\ A 20 lbs increase raised odds to 3.87.

## Odds vs probability

One could have high odds but still low probability for something. You may increase your odds of something but the probability of getting that outcome was still low to begin with. Another may have lower odds but high probability of getting an outcome.

Take the case of people in different ages on different diets and on different drugs and their chances of them getting sick because of it. Younger people have a low probability of getting sick whether or not they do things that increase their odds of getting sick.

Odds can have a large magnitude change even if the underlying probabilities are low.

## Logit

We don't know p and we wish to estimate it. The estimate of p is written src_LANG[headers]{\hat p} (p hat). We need a function that links the independent variable x axis with probabilities on the y axis.

## Regression equation

We are estimating an unknown p for any given linear combo of independent variables. In the logit function we have 0 to 1 running along our x axis but we want to have them on our y axis. We can achieve that by taking the inverse of the logit function.

### Example

Home owners loans n = 1000 1 approved 0 not aprroved

# Finite math

order matters

## Combinations

The number of different ways that r objects can be selected from n objects. If there are n objects, how many different ways can we select groups of size r?

Often said as n choose r, denoted as C(n,r)

Order doesn't matter Think of sets.

### Formula

C(n,r) = \dfrac{n!}{r! (n-r)!}

# Discrete distributions

The outcomes are finite and must be integers

# The Binomial Distribution

A type of discrete distribution.

The probability of any given outcome is a combination of both the number of trials and the success rate.

Binomial Bi two and nomial is a name in our case an outcome, 2 outcomes. We categorize our outcomes as either a success or a failure.

## Binomial experiment

### Characteristics

1. You have to have a fixed number of trails.
2. Trials must be independent - outcome of one trial doesn't affect any other
3. Each trial has only 2 outcomes a success or a failure
4. The probability of success remains the same in every trial

### Formula

Where (look under [[Finite math]]):

• n: number of trails
• x: number of successes in n trials
• p: probability of success in a single trial
• q: probability of a failure in any trial

### Example

In a die, what is the probability of rolling a 4 is 30%. The die is rolled 10 times. Find the probability of rolling eight 4s.

Solution

# Standard Error/Standard Error of the mean

This is the estimated population standard deviation from the sample standard deviation. Sample mean is unlikely to be equal to population mean. Standard deviation of the means of many samples from the population mean.

# ANOVA (ANalysis Of VAriance)

This is the variability among/between sample means vs variability within each sample

\ Therefore, the samples are likely to come from the same population. \ Why not multiple t-tests? The error compounds in each t-test. \

ANOVA is really a variability ratio:

• partitioning :: separating total variance into its component parts

If variance between the means is relatively large than within the means ratio will be much larger than 1 and the samples likely don't come from a common population.

Overview

At least one mean is an outlier and each distribution is narrow; distinct from each other

Means are fairly close to overall mean and/or distributions overlap a bit, hard to distinguish

Means are very close to overall mean and/or distributions melt together

## One way ANOVA

Also called single factor ANOVA (ANalysis Of VAriance).

Without getting the avg of the sum of squared deviations

SST (Sum of Squares Total)

1. Find difference between each data point and overall mean
2. square the difference

SSC (Sum of Squares of the Columns)

1. Difference between each group mean and overall mean
2. Square the deviations

SSE (Sum of Squares Error)

1. Find the difference between each data point and it's own column mean
2. square each deviation

SST = SSE + SSC

Sum of squares: SS = \Sigma (x-\mu)^2

Sample variance:

### Example

H_0: \mu_1 = \mu_2 = \mu_3 \ H_\alpha: There is at least one difference among the means \alpha = 0.05

|---|---|---| | 1 | 2 | 3 | |---|---|---| | 1 | 2 | 2 | | 2 | 4 | 3 | | 5 | 2 | 4 |

Means within:

Means between:

Degrees of freedom:\

• k :: number of conditions
• N :: number of scores

From the above we get the F_{critical} from our table.

For the above in our table we get F_{critical} of 5.14

$$ANOVA = \frac{SS_{total} - SS_{within}}{SS_{within}} = \frac{SS_{between}}{SS_{within}}$$

Therefore, we fail to reject our H_0 \ /Mean squared between/ is also /variance between/

Out of scope

# z-test, t-test & p values

How to conduct hypothesis tests on 2 population means.

## t-test

### t value

Shows the difference within groups and compares it to difference between the same groups. In the case of the paired t-test we get the t value for paired data.

# Matched sample (paired) t-test

The paired t-test - also called two sample, within subjects, repeated measures and dependent samples t-test - is a statistical method used to measure the change within the same sample after an event occurs. It uses paired or dependent data (where the data in one sample affects the data in the other sample e.g before and after a process such as taking a drug).

#### Properties of the paired t test

• holds more statistical power as there isn't variability between subjects
• susceptible to ordering effects
• used where we care about the difference between each observation
• assumes the difference between pairs is normally distributed

#### Example

Paired t-test on the effectiveness of a weight loss drug.

#+CAPTION: Positive values indicate weight loss and negative values indicate weight gain | subject | on drug | on placebo | d_i | |---------|---------|------------|------| | 1 | 1.1 | 0 | 1.1 | | 2 | 1.3 | -0.3 | 1.6 | | 3 | 1.0 | 0.6 | 0.4 | | 4 | 1.7 | 0.3 | 1.4 | | 5 | 1.4 | 0.7 | 0.7 | | 6 | 0.1 | -0.2 | 0.3 | | 7 | 0.5 | 0.6 | -0.1 | | 8 | 1.6 | 0.9 | 0.7 | | 9 | -0.5 | -2.0 | 1.5 | |---------|---------|------------|------| | | | | 7.6 | #+TBLFM: $4=vsum(@2$4..@10\$4)

• H_0 :: The given weight loss drug has no effect on weight loss.
• H_1 :: The given weight loss drug leads to weight loss.
• \alpha :: 0.05 (threshold for whether to accept H_0).
• d_i :: difference between measurements of each subject
• \mu_d :: mean difference between the measurements of each subject, if \mu_d = 0 there's no difference between the two measurements.
• degrees of freedom (df) :: n - 1
##### Conclusion

The result is significant at p < 0.05 Since p-value (0.005) > than \alpha (0.05), we reject H_0. Therefore, we accept H_1 that our drug is effective at weight loss because there's only 0.005 chance that the weight loss was not because of the drug.

## P values

Get them from the p tables given the z score

# Types of variables

Types of variables:

• numerical ::
• derived :: e.g. Body Mass Index
• transformed :: e.g logarithm
• qualitative :: non-numeric
• categorical :: discrete yes or no
• quantitivative :: numeric
• discrete ::
• continuous ::

Exposure and outcome variables

• predictor ::

• response ::

• Cumulative frequency :: summation of frequency

# Hypothesis testing

Testing whether a claim is valid.

# Types of studies

• observational studies
• case control study
• cohort study
• cross sectional study
• controlled experimental study
• before-and-after-type study
• experimental study
• blinded