MATH 2970: Exam 1 (Data Science)

Four Phases of a Statistical Investigation

1) Posing a Question

2) Collecting Data

3) Analyzing Data

4) Interpreting Results

Posing a Question

Consider context questions and explaining variability (tolerant for a variety of answers)

Collecting Data

Consider sampling and measurement

Analyzing Data

Analyze attributes of data.

1) Center — Mean, Median, Mode

2) Dispersion — Range, IQR, MAD

3) Distribution Shape — Shape of Graph

4) Relationship (association) — How closely on variable changes the other (bivariate)

Interpreting Results

Answering questions through data

Why should teachers teach data science?

1) Getting to know your students

2) Every student participates regardless of ability

3) There is no right answer (answers can vary)

4) Data makes learning engaging and interesting

Statistical Questions

Questions in which data can be collected and responses involve data that can vary.

1) There needs to be a specific answer?

2) Is there more than one way to answer the question? (can’t be a fact)

Categorical Data

Nominal — no set order

Ordinal — set order (class/sibling rank or birth order)

Numerical Data

Discrete — Whole #s (integers too)

Continuous — Accurate to a place value (decimal/fraction)

Activities that provide data

Observational Study — gather data and take measurements on changes (observing patients)

Simulation — create a false situation, run scenarios, observe what happens

Survey — choose a subset of population and get input

Experimental Study — set up treatments (sometimes called a placebo) and have control groups

Valid

Means the population is being represented as accurately as possible

Reliable

Means the population data is consistent (repeatable with multiple measurements)

Population

Everyone is included. Ex: Census

Most accurate but never completely accurate because the population changes (DIFFICULT)

Sample

Subset of the population (Survey, Observation, Simulation, Experiment)

Depends on the bias, representation (validty), randomness, and sample size

Picture Graph

Represents data with images or symbols for easier interpretation

Bar Graph

Represents data from different categories for comparison

Line Graph

Shows changes and trends in data over continuous period of time

Dot Plot

Displays a distribution with “dots/points” above a number line to show distribution shape, center, and variability.

Histogram

Represents the distribution of a continuous variable by showing the data from least to greatest within intervals which are sometimes called “bins”. (Has connected bars)

Cut off ←

Circle Graph

Show proportions of a whole, represented as percentages or fractions. (pie chart)

Stem and Leaf

Displays a distribution of numeral data with the look of a bar graph

Box Plot

Shows distribution of a data set using the minimum, Q1, median, Q3, and the maximum. Shows the range and interquartile range and can help identify outliers.

Frequency Table

Chart that organizes data so you know how often each value occurs. (Tally Chart)

Scatter Plot

Shows the relationship between two numerical variables using x and y axis to identify trends, outliers, and possible correlation.

Positive, Negative, or Weak relationship.

Normal Bell

Doesn’t touch bottom (x-axis)

Rectangular (Uniform)

No change in data

J-Shape

Exponential Growth (keeps going)

Reverse J-Shape

Exponential Decay (keeps going)

Right Skew

Tail is on right

Outlier on right

Left Skew

Tail is on left

Outlier on left

Measures of Center

Mean, Median, and Mode

“Typical” means any type of measure of center.

Mean

Average.

Add them all up and divide by number of how many you have.

Do not use if skewed/have outliers. Means goes toward tail.

Relies on numerical data.

Median

Middle — order least to greatest.

Use median in skewed data — more accurate measure of spread for it.

Relies on numerical data.

Mode

Most - # that appears the most.

Often occuring.

Only one that can be used in categorical data.

Measure of Dispersion

Measure of how spread out data is in a numerical set.

“Varies” means any types of dispersion or deviation.

Low Dispersion

Data points cluster tightly around the center.

Predictable, consistent data.

Important: Food and Behavior

High Dispersion

Data points are scattered far from the center.

Volatile, unpredictable data.

Important: Identifying high or low level academic or behaviors

Range

Highest - Lowest

Interquartile Range

Box Plots — Q3 - Q1

Don’t worry about outliers here.

Mean Absolute Deviation

MAD. Mean and the absolute value of how far it is from each value, added all up, and divided by total numbers of numbers.

Don’t use when there’s outliers.

How far the values are from the mean, on average.

X Axis Variable

Input, independent, or predictor variable

Y Axis Variable

Output , dependent, or response variable

Least Squares Line

When relationship is linear in a scatter plot, it can be described with a Graphed Straight Line: Line of Best Fit, Best-Fit Line, Trend Line, Regression Line, or Least Squares Line.

Dots distributed evenly on both sides of the line.

Used to determine relationship: positive (as one variable increases, the other variable also increases), negative (as one variable increased, the other variable decreases), weak or none (there is no clear direction to the points, they appear randomly placed)

Correlation

A measure of how strongly 2 variables move together, showing a pattern, also called an association.

Coefficient of Determination

r

measures the strength and direction of a linear relationship between two quantitative variables.

WEAK: 0 to 0.25

MODERATE: 0.25 to 0.70

STRONG: r > 0.70

Probability

Likelihood that something will happen

Ranges from 0 (no chance) and 1 (the event will definitely happen)

favorable outcomes/ total outcomes = probability

to compute a probability of something not happening, you can subtract the probability from 1

Calculating the outcomes of a lottery without replacement

For example: Power Ball — to win the person’s ticket must match all 5 white balls (1 to 69) and the red powerball (1 to 26).

69 × 68 × 67 × 66 × 65 × 26 = 3.5 × 10^10

1 / 3.5 × 10^10 is the probability of winning powerball jackpot.

Nobody can buy 35 billion tickets — not enough time or money.

Number of people who buy tickets does not change the probability of winning.

Interpreting Percentiles

A ___ percentile means that ___ percentage of the observation are equal or less .

Ex:

Your kid is in the 65th percentile which means they scored better than 65% of the kids who took it.

80th percentile of standardized test, they scored as well or better than 80% of the students who took the same exam.

Percentiles are ordinal — they show rank and compare.

Interpreting Mean and MAD

On average, ___ is ___.

The data values are ___ units away from the mean, on average.