STA 100 Midterm 1

statistic

numerical measure calculated from sample data; a function of our data

parameter

numerical measure calculated from the entire population

n

of a summation, indicates the end of the index

i=1

of a summation, index/iterator representing the start of a data set

x_i

variable/name of list column in excel

percentile

the pth percentile is the value of y where p% of the data values are less than y and (100-p)% of data values are greater, sometimes denoted P_p

sort the data numerically

first step of the general algorithm for any arbitrary percentile

c

of a summation, a constant that can be factored out from the variable

y bar

sum of all values over sample size, represents sample mean

i.e. 1/n times sum of y_i

calculate the index position number

second step of the general algorithm for any arbitrary percentile

use rounding rule

third step of the general algorithm for any arbitrary percentile

I_p = (p/100)(n+1)

index position number formula

y tilde

median, 50th percentile value of y

i.e. middle of sorted data

five number summary

statistics used to compose a boxplot: min, Q1, Q2 (median/y tilde), Q3, max

mode

the most frequently occurring value(s) of y in dataset

Q1

median of the lower half of the ordered data (not including the median)

Q3

median of the upper half of the ordered (not including the median)

range

{max y_i - min y_i}

interquartile range (IQR)

the range of the middle 50% if the data

Q3 - Q1 or P₇₅ - P₂₅

sample variance (s²)

average squared deviation/distance from the mean

NOT used to interpret variance because the units are squared

(1/n-1) Σ (y_i - y bar)² = (1/n-1)[(Σ y_i²) - n(y bar)²]

sample standard deviation (s)

average or typical deviation from the mean, used to interpret standard deviation

frequency

the amount of times an observation occurs

relative frequency

frequency over the sample size

i.e. proprtion, propbability

percent frequency

relative frequency*100%

i.e. (frequency/sample size)*100%

random variable

a variable whose outcome is the result of some random process and do not know the value/outcome in advance

numeric

RV naturally measured as numbers

continuous and discrete

continuous

numerical RV, take on ANY value in an interval (at least theoretically)

ex. weight, volume, time, temperature, blood pressure, % grade

discrete

numerical RV, numbers with natural gaps between values (often integers)

ex. # counts, people, engines, outcomes of a die roll

categorical

RV naturally measured as labels, groups, factors, levels

nominal and ordinal

nominal

categorical RV, no natural ordering or ranking

ex. eye color, cancer site, car brand, sex, race

ordinal

categorical RV, does have natural ordering or ranking

ex. cancer stage, letter grades, low/med/high, taxonomy

Y

denotes the set all possible values of a RV

Y_i

denotes the i^th value of RV Y an itself still random before measuring

y

denotes one specific value of a RV

bar chart

separated bars with an interchangeable scale of frequency, rel. frequency, or percent

represents categorical data

stacked bar chart

grouped frequency bar chart and variables are stacked

stacked relative frequency

grouped percentage bar chart and variables are stacked to sum to 100%

side by side bar chart

grouped bar chart, typically contains nominal AND ordinal data

histogram

values grouped into bins or intervals of equal width with no gaps

can show bins as freq., rel. freq., or percent

represents numerical data

left skewed

left tail of data is longer

mean < media < mode

ex. % grades

right skewed

right tail of data is longer

mode < median < mean

ex. income, wealth, lifespan of humans

sample space

the set of ALL possible outcomes of an experiment, represented by S

event

any subset of the sample space

probability

a number between 0 and 1 (inclusive) representing how likely an event is to occur

P(A)

theoretical probability

assumes we know all possible events, outcome, probabilities

(# times event A occurs)/(# all possible events)

P hat (A)

empirical probability

an estimate of the true probability based on a sample of observed data

(# times event A occured in sample)/(sample size n)

as n approaches infinity, approaches theo. probability

0 ≤ P(A) ≤ 1

probability rule that states if P(A) = 0, the outcome never happens and if P(A) = 1, the outcome always happens

therefore, P(A) cannot be negative or greater than 1

sum of all probabilities

∑n, i=1 P(A_i) = 1, probability rule that states if we have n possible outcomes of an experiment, then the _________________ is 1

1 - P(A)

probability rule that states the event that A does not occur is A^c (read as A complement)

P(A^c) = _______