Download Schaum 1 Concepts

Basic Concepts in Descriptive Statistics
Schaum's Outline of Elements of Statistics I: Descriptive Statistics & Probability
Chapter 1. Functions.
Function (p 10): If two variables are related so that for every permissible specific value x of X
there is associated one and only one specific value y of Y, then Y is a function of X. The domain
of the function is the set of x values that X can assume, the range is the set of y values associated
with the x values, and the rule of association is the function itself.
Functions in statistics (p 11-12): independent/dependent variables and cause/effect.
In the mathematics function y = f(x), y is said to be the dependent variable and x the independent
variable because y depends on x. In the research context the dependent variable is a measurement
variable that has values that to some degree depend on the values of a measurement variable
associated with the cause.
Chapter 2. Measurement scales (p 35-36).
Nominal: unique mutually-exclusive categories (=,  ), meaning that a measured item is = or 
some category. For example, fish being shark, flounder, or trout.
Ordinal: nominal (=,  ) plus ordered (<,>). For example, eggs are small, medium, or large.
Interval: ordinal (=,  and <,>) plus uniform reference units (+,-). For example, degrees Celsius.
Ratio: interval (=,  and <,> and +,-) plus absolute zero making ratios meaningful. For example,
degrees Kelvin where 300 K is twice as hot as 150 K.
Chapter 3. Probabilities for sampling: with and without replacement (p 60).
The probability of drawing an ace from a deck of 52 cards is P(ace) = 4/52 = 1/13, and if the
sampling is done with replacement, the probability of drawing an ace on a second try is also 1/13.
However, if the sampling is without replacement, the probability of drawing the second ace is
P(second ace) = 3/51.
Chapter 4 and 5. Frequency distributions and graphing frequency distributions.
Chapter 6. Measures of central tendency.
Mean or average:  =  x i N (p 140)
i
Median = value that divides an array of ordered values into two equal parts (p 149)
Mode = the measurement that occurs most frequently (p 156)
Chapter 7. Measures of dispersion.
Variance  2 =  ( xi   ) 2 (n  1) (p 188, unbiased estimate) Standard Deviation is  .
i
Normal probability density function (bell shaped curve): 68% of the values lie within one 
from  and 95% within 2  from  (p 197).
Chapter 8. Probability: four interpretations.
Basic concepts: experiment = a process that yields a measurement, trial = identical repetition of an
experiment, outcome = result of a trial (a measurement), event = a possible specific outcome (p
227).
1. Classical: deals with idealizes situations, like the roll of a perfect die on a flawless surface
having equally likely (probabilities of 1/6) outcomes (p 227).
2. Relative frequency: data from experiments are analyzed to obtain the relative frequency of
events (p 229).
3. Set theory: the basis for the mathematical theory of probability (p 230).
4. Subjective: in contrast to the objective determination of probabilities above, here the
probabilities are determined using “personal judgment” or “educated guesses” (p 237).
Chapter 9. Calculating rules and counting rules.
P(A  B) signifies "the probability of event A or event B." If the events are mutually exclusive,
such as, in tossing a die, event A is getting an even number and event B getting a 5, then the
probability of A or B is simply 3/6 + 1/6 = 4/6, that is
P(A  B) = P(A) + P(B)
(Special addition rule, p 258)
When A and B are not mutually exclusive,
P(A  B) = P(A) + P(B) – P(A  B) (General addition rule, p 264)
P(A  B) signifies "the probability of event A and event B." In terms of the outcome space, we're
thinking about the set-theoretic intersection. Relative to a die, suppose event A is getting a face
divisible by 3 (i.e., 3 or 6) and event B is getting an even number. The only outcome satisfying
both is a 6. Thus, the probability that a die toss is even and divisible by 3 is 1/6.
Conditional probability: P(A|B) signifies "the probability of event A, given the occurrence of
event B." Event B is additional information that can narrow the outcome space (the range).
P(A|B) = P(A  B) / P(B) if P(B)  0 (p 259)
P(A  B) = P(B) P(A|B) (General multiplication rule, p 261)
P(A  B) = P(A) P(B) (Special multiplication rule when A and B independent, p 263)
Bayes’ Theorem (also known as Bayes’ Law)
P(A|B) = P(B|A) P(A) / P(B)
(p 270)
Chapter 10. Random variables, probability distributions, cumulative distribution functions.
A random variable is a function having the sample space as its domain, and an association rule that
assigns a real number to each sample point in the sample space. The range is the sample space of
numbers defined by the association rule (p 309).
Example of a random variable: the experiment is flipping a coin twice, the random variable (a
function) is the number of heads on the two flips, the domain S = {HH, HT, TH, TT}, the rule of
association is counting the number of heads, and the range S = {0, 1, 2}.
A discrete random variable has a sample space that is finite or countably infinite (p 310).
A continuous random variable has a sample space that is infinite or not countable (p 310).
Understand discrete and continuous probability distributions.
Expected value of discrete probability distribution: E(X) =  =  xf ( x)
(p 322)
x
Variance of discrete probability distribution: Var(X) =  =
2
 (x  )
2
f ( x)
(p 325)
xf ( x)dx
(p 324)
x
Expected value of continuous probability distribution: E(X) =  =
Variance of continuous probability distribution: Var(X) =  2 =






( x   ) 2 f ( x)dx (p 328)