Sampling & Distribution

Descriptive Stats

Numbers and graphs used to summarize and describe data.

Accurate Inferences (Conclusions) Require:

1. Good experimental design

2. Representatives samples

3. Accurate theories

Central Tendency

Where most of the data is centered or what a “typical” value looks like.

• Mean = average

• Median = middle value

• Mode = most frequent value

Histogram

It shows how often values occur and the shape of the data (e.g., normal, skewed)

• Histogram = continuous data

• Bar graph = categories

Concrete (Local) Element

Things directly observed or calculated from the sample, such as statistics, graphs, and relationships between variables

Abstract (Global) Element

Help us make conclusions about the population; population estimation, hypothesis testing, confidence intervals, etc.

Distribution

How data is distributed (spread out)

• Tells us the range of scores and the frequency (or probability) of those scores

• Normal, skewed, uniform

Probability Density Function (PDF)

The probability of different values occurring in a continuous distribution.

Tells the relationships between the value and the population mean

Uniform Distribution

• Symmetric and all outcomes are equally likely (roll of die)

• Discrete: finite number of outcomes

• Outcomes are bounded (we know the lowest and highest values, can't go under/past)

Binomial Distribution

• Probability of a win/lose outcome in an experiment repeated multiple times

• Trials are all independent

• Two possible outcomes (coin toss, hit or miss the target with a dart)

Normal Distribution

• Symmetric, bell-shaped, continuous data

• Individual scores in a population or sample

• Described by mean and standard deviation

• Positive (right) skewed, Mode < Median < Mean

• Negative (left) skewed, Mean < Median < Mode

Sampling Distribution

A distribution that contains statistics from samples (the mean) instead of individual scores

• Efficient and cheap

• Allow us to make an estimate about the full population

Kurtosis

Measure how much data is found in the tails of a distribution and how peaked the distribution is.

Leptokurtic Distribution

A distribution that is taller, narrower, and more extreme values. Upside down V

Platykurtic Distribution

Shorter and wider, with lighter tails and fewer extreme values. Upside down U

Distribution of Sample Mean (DOSM)

• The mean of DOSM is the same as population

• Distribution of many sample averages used to understand the true population mean.

• Used for confidence intervals & hypothesis testing

Parameter vs Statistic

Parameter (μ)- a numerical value that describes a population.

(Fixed, usually unknown because we can’t measure everyone)

Statistic (x̄) - a numerical value that describes a sample.

(Calculated, used to estimate the parameter)

Theoretical Populations

A population that is assumed to follow a certain distribution shape, with parameters (e.g., mean and standard deviation) based on previous research.

Monte Carlo Sampling

A method that uses repeated random, independent sampling from a probability distribution to estimate population values.

• The more samples taken, the closer the estimate gets to the true value

Example: Rolling a die thousands of times and using the average result to estimate the mean.

Bootstrap Resampling

A method that repeatedly samples with replacement from the original dataset

• Bootstrapping is useful in small samples sizes, more precise estimates of parameters, and make comparisons across groups

• Estimates reliability without collecting new data

Example

Original data:

[2, 4, 6, 8]

One bootstrap sample could be:

[2, 2, 6, 8] or [4, 4, 4, 6]