STAT 200 pre-midterm

what is a numerical vs categorical variable?

categorical = outcomes fall in different categories

numerical = outcomes can be measured on a numerical scale

• numerical variables can be transformed into categorical (ex. age → age range)

• subgroups = levels (ex. category = faculty, levels = science, arts, etc.)

what are the ways to summarize categorical variables?

• frequency / relative frequency tables

• contingency tables (two-way table)

• graphical displays (bar charts, pie charts - based on frequencies)

what are marginal distributions?

• separate variables into separate tables

• determine distribution of each table

what are conditional distributions and what are contingency tables used for?

• set one level as a condition - this is the total used for determining frequency

  • ex. a place of residence for arts students - arts students is the condition

what are the ways to summarize numerical data?

• graphical displays (histograms, stem-and-leaf displays, boxplots)

• shape of distribution of data

• numerical summaries

how are histograms used for numerical data?

• make categories for numerical values

  • find frequencies for numerical values

• advantage - helps us look at shape of distribution

  • modality, symmetry of distribution, presence of outliers

• disadvantage - lose actual data points

what is modality?

• number of peaks - unimodal, bimodal, multimodal

what are the different symmetries of distribution?

• symmetric

• skewed to the right / positively skewed

  • long right tail

• skewed to the left / negatively skewed

  • long left tail

what are the numerical summaries for numerical data?

• measures of center

  • mean, median

• measures of spread

  • variance, standard deviation, interquartile range

• percentiles (quantiles) / quartiles

• 5-number summary

  • minimum, first quartile (Q1), second quartile (Q2), third quartile (Q3), maximum

how are stem-and-leaf displays used for numerical data?

• split data into 2 parts

  • all except last digit of data = stem

  • last digit of data = leaf

• list unique stems

• list leafs in ascending order

• rotating should match histogram shape

how are boxplots used for numerical data?

makes use of 5 number summary

• draw Q1 and Q3 → make box

• find Q2 / median → draw line

• find boundaries for outliers

  • LB = Q1 - 1.5(IQR)

  • UB = Q3 + 1.5(IQR)

• draw boundary lines (whiskers) → line at value closest to boundary that is not an outlier

  • if no outliers → extend whiskers to min and max

• outliers outside of boundaries marked by circles

• draw min and max

what is the mean and how to calculate?

• the average of a dataset

• sum of all observations / number of observations

what is the median and how to calculate?

• exact middle value of a dataset

• if odd number of data points

  • = ((n+1) / 2)th data point

• if even number of data points

  • = average of (n/2)th + (n/2+1)th data points

what is variance and how to calculate?

• shows total variation

• squared deviations of values from the mean

what is standard deviation and how to calculate?

• the square root of variance

• s = sqrt(s²)

what are percentiles / quartiles?

• position where a certain amount of data points are below it

  • quartile 1 = value in data set that has 25% of values below it

  • quartile 2 = 50%

  • quartile 3 = 75%

what is the interquartile range and when is it used?

• different between Q1 and Q3 (Q3 - Q1)

• used when you have skewness or outliers

  • better that using standard deviation / variance for these conditions

how does shifting data affect measures of center / spread?

• add a constant c to each observation in the data

• any measure of center (median / mean) shifts by constant c

• shifting the data does not change the spread (variance, SD, range, IQR)

how does scaling data affect measures of center / spread?

• multiply each observation in the data by a positive constant c

• measures of center and spread will be multiplied by constant c

• variance of the new data will be c² times the original variance

when is standardizing data used?

• to compare observations measured on different scales

  • ex. different currencies

• to compare observations from two different distributions

  • ex. class averages across different semesters

what is a z-score and how to calculate?

• z = observation - mean / SD

• gives the distance between an observation and the mean in units equal to the standard deviation

  • the number of standard deviations that a value is above of below the mean

  • z = 0 → observation = mean

what are characteristics of the normal model?

• bell-shaped, unimodal

• symmetric about the mean 𝜇

• spread of distribution determined by the value of SD 𝝈

• denoted by N(𝜇, 𝝈)

what are terms used for population vs sample standard deviations?

• population numerical summaries = parameters

  • 𝜇 = mean, 𝝈2 = variance, 𝝈 = SD

• sample numerical summaries = statistics

  • ȳ = mean, s2 = variance, s = SD

how are values from the normal model standardized?

• calculate z-score

• z-score follows the standard normal model with mean = 0 and SD = 1

what is the 68-95-99.7% rule?

• Interval → % data falling in interval

• Within 1 SD of mean = ~68%

• Within 2 SD of mean = ~95%

• Within 3 SD of mean = ~99.7%

what is a scatterplot and when is it used?

• helps visualize possible relationships between 2 quantitative variables

• explanatory variable plotted on x-axis

• response variable plotted on y-axis

  • explanatory variable is believed to have influence on the value of the response variable

what are the patterns of a scatterplot that must be described?

• direction

  • positive → x and y values tend to go in the same direction

  • negative → x and y values tend to go in the opposite direction

• form

  • linear vs non-linear

• how scattered are the points?

  • strong relationship → points close to each other

  • weak / no relationship → points spread out / randomly scattered

• any outliers?

  • any points outside of pattern seen

what is correlation and the correlation coefficient?

• correlation refers to the degree of linear association between 2 quantitative variables x and y

• correlation coefficient r is a measure of the strength of a linear association between 2 quantitative variables

what are the different types of correlation

• positive correlation = large values of x are linearly associated with large values of y

  • r = +1 gives perfect positive correlation

• negative correlation = large values of x are linearly associated with small values of y

  • r = -1 gives perfect negative correlation

what are properties of the correlation coefficient r?

• swapping x and y values does not affect the value of r

• the value of r does not change if all values are shifted or scattered;ed

• r is sensitive to outliers, may not give a reliable measure of strength of a linear relationship in the presence of outliers

how do association and causality differ?

• the existence of a linear relationship between 2 variables x and y does not imply that an increase in one variable leads to an increase of decrease in another

  • association does not imply causation

• there may be a lurking variable (third variable) that associates both x and y