Descriptive Statistics

• Descriptive statistics provide simple summaries of data and observations
• Descriptive statistics are used to describe the basic features of data
• The basic features of data provide simple summaries of the observations

Measures of Central Tendency

• These give an idea of where the center of the data lies
• Calculation: Sum of all data points divided by the number of data points
• Formula: Mean = ΣX/N
• Median is the middle value when data points are in ascending or descending order
• If there is an even number of data points, use the average of the two middle numbers
• Mode is the most frequent value in the data set
• There can be more than one mode (bimodal, multimodal) or none at all

Measures of Variability (Spread)

• These describe the spread or dispersion of data
• Range Calculation: Difference between the max and min in the data set
• Range Formula: Range = Max - Min
• Variance measures how far each data point is from the mean
• Variance Calculation: Average of the squared differences from the mean
• Variance Formula: Variance = Σ(Xi−μ)² / N
• Standard Deviation is the square root of the variance
• Standard Deviation represents the average distance that data points are from the mean
• Standard Deviation is easier to interpret than variance because it's in the same units as the data
• Formula: SD = √Variance

Measures of Position

• These describe the position of a particular data point within the dataset
• Percentiles: Indicate the relative standing of a value within a data set
• The pth percentile is the value below which p% of the data falls: 25th percentile (Q1) is the value below which 25% of the data fall
• Quartiles: divide the data into four equal parts
• Q1 (First Quartile): 25% of data falls below
• Q2 (Median): 50% of data falls below
• Q3 (Third Quartile): 75% of data falls below
• Interquartile Range (IQR): The range between the first and third quartiles (Q3 - Q1)
• IQR: Used to measure the spread of the middle 50% of the data

Frequency Distributions

• These are used to summarize data
• These show how often each value or range of values occurs in a dataset
• Frequency Table: Lists the values or intervals and how often each occurs
• Histogram: Graphical representation of the frequency distribution
• Histogram: Data is grouped into bins or intervals
• Bar Chart: A chart representing categorical data with rectangular bars
• Bar Chart: The height/length of each bar represents the frequency of a category

Skewness

• Skewness measures the asymmetry of the data distribution
• Positive Skew: The right tail (larger values) is longer than the left tail (smaller values)
• Negative Skew: The left tail (smaller values) is longer than the right tail (larger values)
• Symmetrical Distribution: The tails on both sides of the mean are roughly equal

Kurtosis

• Kurtosis measures the "tailedness" of the data distribution
• Leptokurtic: Distributions with heavy tails (more extreme outliers)
• Platykurtic: Distributions with light tails (fewer outliers)
• Mesokurtic: Distributions that are normal or close to a bell curve

Visual Representations

• Box Plot: A graph that shows the distribution of data based on the five-number summary (min, Q1, median, Q3, max)
• Dot Plot: Displays individual data points, often used to show small data sets
• Pie Chart: Used to show relative proportions of categories in a dataset

Importance of Descriptive Statistics

• Summarize Data: Helps reduce large volumes of data into understandable forms
• Identify Patterns: Gives insights into trends, patterns, and distributions
• Aid in Decision Making: Descriptive statistics are often used as a first step before conducting inferential statistical tests to guide further analysis

Key Takeaways for Descriptive Statistics

• Descriptive statistics summarize and organize data to provide insights into its overall structure, trends, and variation
• Descriptive Statistics don't make predictions or generalizations beyond data at hand
• Descriptive Statistics are essential prior to understanding data before more complex analysis

Median and Main

• Knowing both the median and the mean is important because they provide different insights into the central tendency of a dataset
• Each measure has its own advantages, depending on the type of data and the specific context

Mean (Average)

• Calculation: The sum of all the values in the dataset divided by the number of values
• Best Used When: The data is normally distributed (follows a bell-shaped curve) and there are no extreme outliers
• Advantages: Uses all data points, mathematically efficient and useful in statistical tests and models
• Limitations: Can be influenced by outliers or extreme values (like [1, 2, 3, 1000], skewed by 1000, not typical)

Median

• The middle value when the data points are arranged in ascending/descending order
• If there is an even number of data points, the median is the average of the two middle values
• Best Used When: there are outliers or is skewed (i.e., the distribution is not symmetric)
• Advantages: resistant to outliers and skewed distributions, dealing with skewed data/outliers

Limitations with the Median

• Doesn't use all data points, so it might not be as informative in datasets that are symmetrically distributed/no outliers

The necessity for both Mean and Median

• If data is symmetric (e.g., bell-shaped/normal), both the measures will be the same or similar
• The mean provides a very good measure of central tendency for symetrical data
• If the data is skewed (e.g., income distribution where a few people earn very high incomes), the mean will be pulled in direction of the skew
• The median will better represent the "typical" value of the dataset
• When you have both the mean and median, you can get a sense of the shape of the data distribution

Understanding distributions

• If the mean > median then the Data is positively skewed (right tail is longer)
• If the mean < median then the Data is negatively skewed (left tail is longer)
• If the mean ≈ median then the Data is roughly symmetric or normally distributed

Handling Outliers

• Reliance on the mean is misleading if outliers are present
• The Median is not affected by outliers and might show a better sense of the central distribution

Summary For Mean and Median

• Mean is useful for symmetric distributions without extreme metrics
• Median is better for skewed data when you want to avoid the influence of outliers
• Knowing both of each of these metrics allows you to better understand and perform more accurate decisions

P-Value

• A p-value is a probability that helps determine whether the results of your statistical test are statistically significant
• It tells you how likely the observed results occurred by chance under the assumption that there is no true effect or relationship (the null hypothesis)

Null Hypothesis (H₀)

• Before conducting the test, start with the null hypothesis
• This is used when there is no effect or no difference in the population

Conduct Test

• Perform your statistical test (like a t-test, chi-squared test etc.)
• Calculates the p-value based on your data

p-value Interpretation

• The p-value is the probability of obtaining results at least as extreme as the ones observed, assuming the null hypothesis is true

Small p-value

• Suggests the observed data is unlikely under the null hypothesis
• Implies that the null hypothesis may not be true
• You might reject the null hypothesis and conclude that is a significant effect or relationship

Large p-value

• Data is likely under the null hypothesis and rejectible to the null hypothesis
• Conclude that there is no significant effects or relationships

Common Thresholds

• p < 0.05 is statistically significant and indicates theres strong evidence to reject the null hypothosis
• p > 0.05 suggests that the evidence is not strong enought to reject the null and there isn't enough evidence to make significant effect

Example

• To test if a new drug is more effective than a placebo, conduct a t-test and get a p-value of 0.03
• This means there is a 3% chance that the observed difference in effectiveness between the drug and placebo is due to random variation (under the assumption that the drug has no real effect)
• Since 0.03 is less than the 0.05 threshold, reject the null hypothesis
• conclude the drug is likely to be more effective than a placebo

Important P-Value Notes

• p-value does not tell you the size or importance of the effect
• P-value indicates whether it is statistically significant or not
• it doesn't confirm that the null hypothesis
• It only suggests insufficient evidence to reject it

Alpha Value

• Alpha value (α) is the threshold you set to see if your results are statistically significant
• Represents making a Type 1 error(incorrectly reject the null hypothesis)

Simple Terms

• Alpha is a cutoff that you compare your p-value against to say whether or not to reject the null hypothesis
• It's the level of statistical error that you are accepting

Common Alpha Values

• α = 0.05: willing to accept a 5% change of incorrectly rejecting the null hypothesis
• α = 0.01: willing to accept a 1% change of incorrectly rejecting the null hypothesis
• α = 0.10: willing to accept a 10% change of incorrectly rejecting the null hypothesis

How the Alpha Value works

• If p-value < α then you reject the null hypothesis as it's statistically significant
• if p-value ≥ α then you fail to reject the null hypothesis as it's statistically significant

Alpha Representation

• alpha (α) indicates if the null is true, you are still willing to accept a 5% chance of coming up with a significant effect when there isnt one

Key Takeaways

• The alpha value determines your tolerance or risk to type 1 errors
• Sets the alpha value which is set before tests
• compares the p-value to decide whether to reject the null hypothesis.

Confidence Intervals

• A confidence interval (CI) is a range of values used to estimate a population parameter
• the parameter in question is anything such as the mean or proportion, based on sample data
• It provides a measure of uncertainty around the estimate
• CIs show confidences of true population value

Confidence Interval Form

• Usually expressed as : Estimate ± Margin of Error
• or as: (Lower Bound, Upper Bound)
• 95% CI Example (150g, 160g): 95% confident that the true average weight of all apples lies in that bounds

Significance of Confidence Intervals

• Provide a range instead of a single value, accounts for sampling variability and creates a more realistic analysis
• Indicates precise estimates
• Help in Decision-Making with effective analysis strategy

Visual representation of CIs

• Overlapping significantly of two CI can indicate to be no strong difference between groups
• If it doesn't overlap then its stronger

Interpreting Common Confidence Intervals

• 90 % CI → has a 90 % chance thats within the interval
• 95 % CI (most common) → has a 95 % chance thats within the interval
• 99 % CI → has a 99 % chance thats within the interval

Common Misconceptions

• DOESNT mean there is a % chance the parameter is between the interval
• It means to say; if we took samples and CIs, % would contain the true parameter

How Cis Are Calculated

• For a mean: x̄ ± (z* × σ / √n)
• For a proportion: p ± (z* × √(p(1-p) / n))

CI Calculations

• study's height of school is x̄ = 165 cm
• sample = 50
• deviation = 10 cm
• Confidence level = 95% (so z* = 1.96)
• error margin is : 1.96 × 10 / √50 = 2.76
• 95% level is ≈ 165±2.76 ≈ (162.24, 167.76)

CI Take Away

• offer a range instead of being precise
• Helps with precision and reliability
• Help in decision-making
• Large means more uncertain, when small means precise

Inferential Stats Tests

• Important values/results below show summaries of when and how they tests should be best used

T-Test (Unpaired)

• Commpares means if two groups are independent/seperate
• ex : treatment vs control group
• Key is P value, stats
• look at interval diff or free
• 0. 05 : Significant is in group
• 1. 05 : No significant

T-Test (Paired)

• When both groups are at two different instances
• value or stats and degrees of freedom
• 0. 05 : Significant within
• 1. 05 : No significant within

Chi-Squared

• Looks at seperate metrics compared to what exepcted
• p value, chi squaared stats,df
• 0. 05 or significant value

F-Tests

• Compares between 2 + groups
• p value,F stats, or degs of freedom
• is significant

Regression

• Looks at relations in variance
• p value ,coefficient factor + CI
• Value can indicate if predictors can be impactful

Notes

• all are p values that have chance of showing outcomes assuming null is correct
• CI , the values show if results are to happen
• If the regression indicates the variance then suggest better model

T-Tests More in Depth (Unpaired):

• See if there are differnces in between two groups
• See if the groups are statistically different or not
• statistic: difference means and group is higher
• (df): values are independent
• 0. 05: then the relationship in group is good

T-Tests More in Depth (Paired):

• Checks when the mean are different or the same at differing times after another
• P Value
• and t-statistics and degrees of freedom are calculated but in comparison in the group:
• 0. 05 change occur

Chi-Squared More in Depth

• sees to asses if the two has significant difference in metrics
• Checks if the difference from values compare and how related

•  stats show values from what is good to expect
  

ANOVA & T-Tests Checks

• The group of the study requires shaprio wilk test results or histogram

Data Test Guide :

• How to make a decision of what test one is choosing

Independent metrics :

• Type of questions :*
• is the dependent variable
• How many groups are compared
• are the gorups related
• Is there much impact

Corelated variables :

• Check normality using histogram results from sharpiro wilk tester (anova vs ttests)
• Use scatter plots

Key takeaways

• Ttests is for grouping
• square for categorize
• Regression is to depict realtionship

What Is Your Dependent Varable(DV)?

• Numerical - GOTO 2
• Categorical - GTOTOS 5

HOw Manny Groups

• 2- GO2 3
• More than 2 - GO TO 4 regression

Are 2 groups connected / indepedent?

• Indepoendent use unpaired t test
• Paired groups (same group mesuared twce)

More than two Groups AnOva

3+ groups do one way anova
   Use repeat

5

Is Depentent Catagorize?? comparing Use the chi square

additonal consider

check vaue diff test data not normal - test signed rank mulitples impacr mulitply

Correlation Is In relation to Causation

Biggest issue is to not to assume the relations

Definition Looks at direction in 2 variables

Features Correlation use factors in between -1 / +1 1 is positive 1 is negative 0 means they dont move together Examoke" ICE CREAM AND DROWN

Cause One variable with effect others

Features cause happens through things and process mostly used in lab setting

Example smoking lungs

Correlation Not Cause they move together

Varrable Un Seen variable more fire fight = more demage Reverse Issue - which effctting what issue- strees make you not slpee or vice versa - no connection

• the number drowned and more are related because the number of people and movies Nic starred a year.

Test Causation experiment with long term tests and regression

Suumary Of Correlation Issue?

• they move together ,where ause move in direction with one
• Statistic vs expereiments
• causation shows direction
• cause smoking results cancer

Take Always cause doesnt mean it relates

variables can exist relationship to test you we need studies statisstics