Chapter 1-4: Unit 1 Elementary Statistics

𝜇

population mean

𝜎

population standard deviation

𝑠

standard deviation

𝑛

sample size - the number of data values in a sample

N

population size

𝜎^2

population variance

s^2

sample variance

𝑥̅

sample mean

x

variable

parameter

numerical summary of a population

statistic

numerical summary of a sample

Qualitative Data

the result of categorizing or describing attributes or characteristics of a population Ex: gender, zip code, color of eyes, hometown

Quantitative Data

the result of counting or measuring attributes of a population. Ex: height, weight, GPA, time.

Continuous Data

Data that can take on any value. There is no space between data values for a given domain. Graphs are represented by solid lines.

Discrete Data

Data with either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point

Convenience Sampling

sample in which the researcher uses data that was easy to get

Simple Random Sampling

every member of the population has an equal probability of being selected for the sample

stratified sampling

sample in which the researcher subdivides the population into at least two different subgroups (or strata), and then draws a sample from each subgroup. "Some from all"

systematic sampling

select some starting point and then select every kth element in the population

cluster sampling

sample in which the researcher first divides the population into sections (or clusters), and then randomly selects all members from some of those clusters. "All from some"

representative sample

a sample that accurately reflects the characteristics of the population as a whole

sampling bias

A problem that occurs when a sample is not representative of the population from which it is drawn; tends to favor certain results

response bias

exists when the answers on a survey do not reflect the true feelings of the respondent. (Participants may lie)

non-response bias

exists when individuals selected to be in the sample who do not respond to the survey have different opinions from those who do. (Low response rate)

z-score

• It represents the number of standard deviations a data point is above or below the mean.

• If it is positive, then it is above the mean.

• If it is negative, then it is below the mean.

• It is a standardized measurement since it is in terms of standard deviation.

• The z-score has no unit of measurement.

Measures of Center

mean, median, mode

mean

(average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set.

median

the middle value when a data set is ordered from least to greatest.

mode

the number that occurs most often in a data set.

Σ

Sum of a set of data value

𝜒

Variable used to represent the individual data values

𝜇

"mu" the mean of a set of population values

𝑥̅

the mean of a set of sample values

measures of location

z-score, quartile, percentile

measures of dispersion/spread

standard deviation, variance, interquartile range, range

5 number summary

a subset of the data that consists of the minimum value, the first

quartile, the median, the third quartile, and the maximum value.

outlier

a data point that is not consistent with the bulk of the data from that group.

Bell-shaped Distribution

the highest frequency occurs in the middle and frequencies tail off to the left and the right of the middle

Right-Skewed (positively skewed)

The tail extends to the right of the peak longer than to the

left. There are extreme values (outliers) to the right.

Left-Skewed (negatively skewed)

The tail extends to the left of the peak longer than to the

right. There are extreme values (outliers) to the left.

Standard Deviation (SD)

Measure of how data deviates from the mean.

standard deviation..

can never be negative. It is positive or zero.

It is zero if all the values are exactly the same.

• Larger values of standard deviation indicate greater amounts of variation.

• Outlier(s) can drastically change the value of the standard deviation since it is a "not resistant" measure.

Relationship between standard deviation and variance

variance = (𝐬𝐭𝐚𝐧𝐝𝐚𝐫𝐝 𝐝𝐞𝐯𝐢𝐚𝐭𝐢𝐨𝐧)𝟐

standard deviation = √𝐯𝐚𝐫𝐢𝐚𝐧𝐜𝐞

measures of center

µ, 𝒙, mean, median, mode

Measures of spread

variance, standard deviation, range, IQR, σ, 𝝈𝟐, 𝒔𝟐, s

complement

all outcomes in which the event does not occur

multiplication rule

the probability of two events both happening

The Multiplication Rule for Independent Events

If events A and B are independent, then 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵).

The addition rule

the probability of either one of two events happening, 𝑃(𝐴 𝑜𝑟 𝐵)is

equal to 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)

The Addition Rule for Mutually Exclusive Events:

If events A and B are mutually exclusive, then 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵).

Mutually exclusive

events cannot occur at the same time, meaning that they share

no common outcomes and 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0

binomial distribution

sequence of two outcomes

Properties of a Binomial Distribution

• Two mutually exclusive outcomes ( success and failure)

• Fixed number of independent trials

• P(success) remains constant from trial to trial

Notation for Binomial Experiments

n = number of triasl

p = the probability of success (percentage)

q = the probability of failure ( 1 - p )

x = random variable

𝑝

𝑃(𝑆) The probability of success in a single trial

𝑞

𝑃(𝐹) The probability of failure in a single trial

𝑞 = (1 − 𝑝)

Expected Value (Mean of the probability distribution of a discrete random variable)

• Weighted average of all possible values over the sample space

• 𝜇 = ∑(𝑥 ∗ 𝑓(𝑥))

Variance of a discrete random variable

• Weighted average of squared deviation about the mean

• 𝜎2 = ∑[𝑥2 ∗ 𝑓(𝑥)] − 𝜇2

Standard deviation of a discrete random variable

𝜎 = √∑[𝑥2 ∗ 𝑓(𝑥)] − 𝜇2