Exploring Data

descriptive methods

the different methods for organizing and summarizing collected data, including tabular methods, graphical methods, and numerical methods

categorical/qualitative variables

variables that place the individuals being studied into one of several groups or categories

numerical/quantitative variables

variables that have outcomes that can be analysed using arithmetic operations

univariate data

data with only one measurement on each object

bivariate data

data with two measurements on each object

frequency

the number of times that an observation occurs, usually denoted with f

relative frequency

the ratio of the frequency (f) to the total number of observations (n), usually denoted by rf, and rf = f/n

cumulative frequency

the number of observations less than or equal to a specified value, usually denoted by cf

frequency distribution table

a table giving all possible values of a variables and their frequencies

center of a distribution

the “typical” or central data point, measured in several ways, including mean, median, and mode

spread of a distribution

how far the data points are from the center, measured through the range, standard deviation, or variance

shape of a distribution

tells where most of the data is, can be symmetric or skewed

symmetric distribution

when the left half of the distribution is approximately a mirror image of the right half, meaning that the data is spread out in the same way on both sides, with the same amount of data on both sides of the center

skewed distribution

when there are extreme values in only one direction that causes one side to have a longer tail, being right-skewed if the tail is on the right, and left-skewed if the tail is on the left

outliers

an observation that is surprisingly different from the rest of the data

stem in stemplot

the left-most part of each observation

leaf in stemplot

the remaining part of each observation, excluding the left-most part

percentage frequency/relative frequency

the frequency of an observation in relativity to the whole sample

population

the entire group of individuals or things

sample

the part of the population that is studied

mean

the average value in a data set. nonresistant and affected by extreme or outlier measurements. for a population, denoted by μ, and for a sample, denoted by x̄

median

the point that divides the measurements in half. resistant and not affected by extreme or outlier measurements, better to use for skewed data or data sets with outliers. sometimes denoted as M

range

the difference between the largest and smallest measurement in a data set, not reliable as it depends on the two extreme measurements

interquartile range (IQR)

the range of the middle 50% of the data, or the difference between the third and first quartiles. resistant and not affected by extreme or outlier measurements

standard deviation

a measure of variation that takes every measurement into account. nonresistant and affected by extreme or outlier measurements

variance

the square of the standard deviation

percentiles

the division of a set of values into 100 equal parts

quartiles

the division of a set of values into four equal parts by using the 25th, 50th, and 75th percentiles

standardized scores/z-scores

(Observed measurement - mean) / standard deviation

linear regression

a model to measure the strength of the relationship between two quantitative variables with a linear relation

Pearson’s correlation coefficient

a numerical summary measure calculated to represent the linear dependence of two variables between -1 and 1. the further away from 0, the stronger the relationship

scatterplot

a graphical summary measure used to describe the nature, degree, and direction of the relation between two variables x and y, where (x, y) gives a pair of measurements

linear regression model equation

Y = α + βX where Y is the response variable, X is the explanatory variable, α is the y-intercept, and β is the slope

predicted value of y

ŷ = a + bx

least-squares regression line

a line that minimizes the sum of the squares of the residuals, otherwise known as the line of best fit. the line will always pass through the point (X̄, Ȳ) and will always have the slope β₁ = (r) [S_y/S_x]

coefficient of determination

measures the percent of variation in Y-values explained by the linear relation between X- and Y-values. denoted by R², which is equal to the square of the correlation coefficient. always between 0 and 1.

random error

a measure of how wrong the predicted values were from the measured values - denoted with ε

influential observation

an observation that strongly affects a statistic

residual plot

a plot of residuals versus the predicted values of Y

transformation

a change made to the equation for variables to make a linear form

log transformation

Z = ln(Y) used to linearize the regression model when the relationship between Y and X suggests a model with a consistently increasing slope

square root transformation

Z = √Y = Y^1/2 used to linearize the regression model when the spread of observations increases with the mean

reciprocal transformation

Z = 1/Y¹ used to minimize the effect of large values of X

square transformation

Z = Y² used when the slope of the relation consistently decreases as the independent variable increases

power transformation

ln(Y) and ln(X) used if the relation between dependent and independent variables is modeled by Y = aX^b

contingency table

a table of data classified by r categories of classification criteria 1 and c categories of classification criteria 2

marginal frequency

the frequency with which each category occurs

conditional relative frequency

the relative frequency of one category given the other category has occurred

association

a measurement of relation between two categorical variables