Chapter 2 - Distributions of Scores Flashcards
Is 80% a good grade?
relative
what is a distribution?
- Conveys relative
frequency with which
values of a variable occur
in a sample or population - Summarize how often
scores occur in a data set - Can be conveyed in tables,
histograms, or polygons
what is relative frequency
relative frequency = frequency of event/total # of event
what is the difference between histograms and bar graphs?
bar graphs = bars don’t touch
histograms = bars touch themselves (continuous)
can you calculate an average for a qualitative variable?
no!
what are frequency tables?
Conveys the number or
proportion of scores in a sample or
population having each value of a variable
define variable, frequency and proportion
what are bar graphs?
Graphical depiction of the
information presented in a frequency
table
* Each value represented by a bar,
heigh represents number or
proportion of scores having that value
* X-Axis shows the area of
preference
* Y-Axis shows proportion of
students choosing each of the
five areas
what is the difference between discrete quantitative variables and continuous quantitative variables?
- Discrete quantitative variables
are typically integers - E.g., There are 31 days in January ( 31 )
- Continuous quantitative variables
are typically real numbers - E.g., The class average o
give an example of a frequency table
- Example:
- We have data from a pop quiz
last term
Pop Quiz Data
6 7 9 8 8 7 7 7 7 6
6 6 7 8 6 7 7 8 4 7
8 7 6 7 7 6 8 8
5 7 10 8 8 7 10 6
7 7 7 6 - Our sample consists of 80 students ( n = 80 )
5 7 * The pop quiz consisted of 10
multiple choice questions - Grades on the quiz ranged
from 0 to 10
what are the titles we look for in a frequency table?
value
f = frequency
cumulative f
p = proportion
P = cumulative proportion
how do we find the value?
value is just the different grades obtained by class
how do we obtain the frequency?
count the number of times a value was received by the students
how do we obtain cumulative frequency?
the number of scores at
or below a given
value of a variable
slide 12 to see how to calculate
how do we obtain proportion?
Divide by the number
of scores in the data
set (n)
how do we obtain cumulative proportion?
the proportion of scores
at or below a given
value of a variable
* 26 / 80 = 0.325 = 0.33
define percentile rank
Cumulative
proportion multiplied
by 100
* 0.325 X 100 = 32.5%
in summary, what are the 5 steps to creating frequency tables?
- Determine maximum/
minimum scores - Count the instances of
each score (f) - Determine the cumulative
frequencies - Determine the proportion (p) of scores by dividing the f by number of scores
- Determine the cumulative
proportion
what is a histogram?
is a graphical depiction of the number or proportion of scores in
a set
* Different than bar graphs:
1. X-axis os placed
in its natural
ordering
2. There is no space
between the bars
What problems might we encounter if we tried to make a frequency table for this data? (grouped frequency tables with discrete-continuous variables)
It would be
enormous, with most
empty as there are
only 60 scores in this
table
so, when should we use grouped frequency tables?
when we have a large
number of possible scores
what are the first and second steps to making a group frequency tables? describe in details.
determine the range of scores and intervals.
First decision is about the number
of intervals and their width
* Must be the same width
* Width should be an integer
* Number of intervals depends
on the number of scores
(typically 5-20)
* Interval width depends on the
number of intervals and the
range of scores
* Range = maximum - minimum
*Range = 96.9 - 38.9 = 58
Grouped Frequency Distribution of 60 Final Grades
* Divide the range by the
number of intervals, e.g., if
we have 10 intervals, 58/10 = 5.8
* Because our interval width
ought to be an integer
(whole number), round up to 6
* Interval width ought to be
intuitive, 5 or 10 versus 6 or 9
what is the third step to making a group frequency tables? describe in details.
what is the fourth step to making a group frequency tables? describe in details.
what is the difference between subjective and objective probability?
- Subjective Probability expresses
individual’s subjective judgement
about the likelihood of something
occurring: “I’ll probably do well on the
exam!” - Objective Probability expresses the
numerical likelihood of some event
occurring (say flipping a coin: 50/50 chance)
is flipping a coin a qualitative or quantitative variable?
qualitative variable: heads or tails
define sampling experiment or trial
every time you flip the coin
what is the proportion of successes?
Nsucesses/Nse (sampling experiments)
A die has 6 sides
* If we want to count the
number of time I roll a one ( 1 )
out of sixty ( 60 ) trials, how
would I do this?
* If I roll a one six times
p = 6/60 = 0.1
how is probability calculated?
by proportion (number between 0 and 1)
A die has 6 sides
* If we want to count the
number of time I roll a one ( 1 )
out of sixty ( 60 ) trials, how
would I do this?
* If I roll a one six times
can a proportion be negative?
no!
define an event
- An Event is one or more of the
possible outcomes of a sampling
experiment - If we said the success in the last
example was rolling a 3 or a 6,
then the event is a 3 or a 6 - We can compute the proportion of
times an event occurs in the same
way we computed the proportion of
time an outcome occurs, e.g.: - 75 rolls, fourteen 3s and sixteen 6s
- p = (14 +16) / 75 = 0.4
what is the probability of an event?
the proportion of time the event would occur if the same sampling experiment were repeated infinitely many times
true or false: “The probability of an event
is the proportion of times
the even would occur in an
infinite number of identical”
sampling experiments”
true!
why should we care about statistics?
- Probability intimately related to our use
of distributions in statistics - Given any distribution of scores, we can define a number of events
- “Score is greater than x”
- “Score is less than x”
- “Score is between x and y”
- “Score is outside the interval x to y”
LAWS OF PROBABILITY
1. if the probability of an event is 1.00
2. if the probability of an event is 0.00
3. if the probability exceeds the values between 0 and 1
4. the sum must be equal to X?
- the event MUST occur
- the event will NEVER occur
- not possible
- 1.00
describe the OR rule
If you are asked the probability
of x OR y occurring, you add
the probability together
describe the AND rule
if you are asked the probability of x AND y occurring, you multiply the probabilities
define mutually exclusive events
cannot co-occur: a coin can come up heads or tails, but NOT both
define independent events
occurrence of one event
does not affect the probability of the other
* If two coins are flipped and one comes up heads it has no affect on the results of the other coin
define dependent events
occurrence of one event does affect the probability of the other
- drawing from a deck of cards without replacement (/52, /52, etc.)
What is the probability of drawing
a heart OR face card in a single
draw from the deck? (what rule do we use: OR or AND?)
OR RULE
Three heart cards also have faces,
these events are not mutually
exclusive, so we need to remove
the cards we counted twice
pHeart = ( 13/52 )
pFace = ( 12/52 )
pHeart&Face = ( 3/52 )
22 / 52 = .42
What is the probability of having
two baby boys in a row? (which rule do we use OR or AND?
AND RULE
pBaby1 = ( 1/2 )
pBaby2 = ( 1/2 )
( 1 / 2 ) * ( 1 / 2 ) = ( 1 / 4 ) = .25
.5 * .5 = .25
- What is the probability of drawing a
queen and then a king from the same
deck of cards? WITH REPLACEMENT
If the queen is put back into the deck,
then on 2nd draw there will still be 52
cards in the deck
pQueen = ( 4/52 )
pKing = ( 4/52 )
( 4/52 ) * ( 4/52 ) = .005917
What is the probability of drawing a
queen and then a king from the same
deck of cards? WITHOUT REPLACEMENT
If the queen is not put back into the
deck, then on 2nd draw there will now
only be 51 cards in the deck
pQueen = ( 4/52 )
pKing = ( 4/51 )
( 4/52 ) * ( 4/51 ) = .006033
define probability distribution
conveys the probability that a
randomly selected score will have a given value or fall in a given interval
why should we care about probabilities?
- Probability intimately related to our use
of distributions in statistics - Given any distribution of scores, we can
define a number of events - “Score is greater than x”
- “Score is less than x”
- “Score is between x and y”
- “Score is outside the interval x to y”
what is a probability density function?
- Probability Density Functions: plots
the density of scores at each value of
a continuous variable - This shows a hypothetical distribution
of heights measured in inches. - There are more scores around the
centre of the distribution - The number of scores decreases
as we move away from the
center.
define density
- Density is the proportion of scores in an
interval divided by the interval width - d = p / w
- p is the proportion of scores
- w is the width
- For each value of x, there is a single
density - The area under the curve is equal to 1
