BEM 201 Test 1

Statistics

A way to transform data into information. Creating new understanding from a set of numbers.

Descriptive Statistics

Summarizes and describes the variables in a dataset using numeric and visual techniques.

Inferential Statistics

Uses sample data to make predictions or decisions about a larger population.

ex. estimation, hypothesis testing, regression. Make generalizations or predictions.

Population

The entire set of entities that we wish to study.

Sample

A subset of the population

Parameter

A descriptive measure of a population. A fixed measure that will describe the entire population.

So given the variable “has blonde hair” which can either be true or false. The parameter would be the total number of people in the population that have blonde hair.

Statistic

A descriptive measure of a sample. It’s valuye depends on which sample is selected. For example we grab a sample size n from the larger population of N people.

Then a statistic would be that x people out of a sample size n have blonde hair.

Confidence level

The proportion of times taht an estimating procedure will be correct.

ex. 95% confidence level means estimates based on this form of inference will be correct 95% of the time

Significance level

Measures how frequently the conclusion will be wrong in the long run.

1 - confidence level

Sampling plan

Is a method or procedure for specifying how a sample will be taken from a population.

1. define population

2. the sampling method

3. Sample size

Simple random sampling

Given a population of N people, each individual in the population has an equal weighted chance of being selected into our sample. The issue is that we would need a complete list of the entire population.

Replacement

Replacement: when doing simple random sampling we add the subject back to the population so they could be selected again

Non-replacement: we remove them from the pool so they could not be selected again

Stratified random sampling

Here the population is divided up into stratum, s.t. the population is grouped. Then we go into each one of these groups and we conduct random sampling within each of the stratum

Systematic random sampling

1. Identify the size of the population ex. N

2. Identiify sample size n, you wish to get to

3. k = N/n, which will tell you your sampling interval\

4. Then choose a random starting point i between 1 and k, 1<=i<=k

5. Include the units at positions i, i + k, i + 2k…

Cluster sampling

The population is divided into naturally occurring groups, called clusters. A random sample of these clusters is chosen, and every (or some) members of each selected cluster is studied.

For example, if a population has 20 clusters, a researcher might randomly select 10 of them and collect data from all members of those 10 clusters.

Convenience sampling

Selecting people from the total population to be in the sample where the data is easy to collect. However this is dangerous and can lead toward conveince bias.

Sampling error

The difference between the parameter and the statistic

Non-sampling error

Non-sampling error is any error that isn't caused by random chance in choosing the sample. It comes from how the data is collected, recorded, or processed.

Observation / Variable

Observations (also called cases or individuals) are the entities being studied, such as people, animals, or companies. Each observation is one row in the data table.

Variables are the characteristics recorded about each observation, such as age, height, or favorite color. Their values can differ from one observation to the next. Each variable is one column.

Cross-sectional data

Cross-sectional data is collected on many entities at one point in time, with each entity measured once.

Example: the scores from one class exam. Every student is measured once on the same test, so the data shows how students compare at that moment.

Time-series data

Time series data is collected over time, with each row representing a time period (day, month, year, etc.) and the variables measured at each one. It's used to track changes, spot trends, and forecast future values.

It often follows a single entity, but it can include more than one.

Example: the daily closing prices of Apple and Microsoft over the past year, with one row per day and one column per stock.

Time series data vs. cross-sectional data

Time series vs. cross-sectional: In time series data, each row is a time period, and the goal is to see how variables change over time. In cross-sectional data, each row is an entity (a person, company, etc.), and all entities are measured at roughly the same moment.

It doesn't have to be the exact same second, but time isn't a factor. The goal is to compare entities, not track change.

Qualitative Data

Two categories

Nominal data: Catagories with no inherent order ex. (is X, vs. is Y) (true false type)

Ordinal data: Catagories with an order ex. a course rating where it is a scale

Quantitative Data

Interval data: equal spacing between values, but zero is arbitrary and doesn't mean "none." Differences make sense, but ratios don't. Ex: temperature in °F or °C, calendar years.

Ratio data: equal spacing and a true zero that means a complete absence. Differences and ratios both make sense. Ex: weight, height, income, age.

Continous vs. Discrete Data

Continous: Can take any value within a range

Discrete: Can onyl take specific seperate valyes, usually whole numbers ex. a count

Relative frequency

frequency of the category / n

Lets say that you have 3 categories, then the sum of all of the relative frequencies should add up to one

Bar chart

Used for nonimal data, where the order does not matter. It descirbes the count, or the frequency for which some variable appears in the data set for each category.P

Pie chart

Also used for nominal data where the order does not matter. Describes the relative frequency for each category and shows them adding up to 1 whole “pie”

Grouped frequency distribution

Displays the number of observations in each of the distributions distinct classes. Used for continous data or when there are many discrete values.

Use a histogram

How to make a histogram

1. determine the number of classes using k >= log2n

2. bin width = (max - min) / k

3. start at the min value in the dataset and count all of the values that belong to that bin

Histogram

A type of bar graph that shows the distribution of a set of numerical data. Counts the frequency for data that belongs in differnt bins / classes

A histogram is symmetric when a vertical line down the middle splits the two sides into identical shape and size. (No skew)

Skewness

When there is a “tail” on one side of the histogram.

Postive / right - A bulk of the observations are on the left side

Negative / left - A bulk of the observations are on the right side

Modality

A mode is the class / value which is observeed the most times in the dataset.

In a histogram modality describes the peaks. A histogram can have multiple peaks and be bimodal (a special case of multimodal)

Joint freq distribution / relative freq

Shows the relationship between two or more variables by counting how many observations fall into each combination of categories or intervals

• summarizes how many observations falls into a catergory

• summarizes how many observations fall into a class / interval
  
  

We can also represent these observations as a percentage of the grand total which would be a joint relative frequency table

measures of center

mean: population and sample mean

population = $\mu$

sample = $\bar{x}$

The median has two cases, either odd values, in which case you do (n + 1) / 2

Or even values where you find average between n / 2 and (n / 2) + 1

mode

the value or category that occurs most frequently

right vs. left skew

median < mean (right skew)

median > mean (left skew)

The extreme values that are included in the calculation of the mean will bring the tal toward the mean

range

largest observation - smallest observation

Variance

sigma ^ 2 = sum (xi - u) / N (population)

sigma ^ 2 = sum (xi - x bar) / n -1 (sample)

Standard deviation

How far on average a value deviates from the mean it is sqrt( sigma ^ 2)