Download Statistical Measures

Statistical Measures
Mrs. Watkins
AP Statistics
Chapters 5,6
MEASURES OF CENTER
Mean: arithmetic average of all data values
population mean:
sample mean:
Formula:
Mode: the most common value in a data set
Median: the middle value in a data set
Midrange: average of the extremes
Trimmed Mean: when you find the mean
of data set with a certain percentage of
data values trimmed of the ends of the
distribution
Ex:
5 number summary
5 important numbers in data set:
Min:
Q1:
Med:
Q3:
Max:
Q1, Med, Q3, may not be actual data values
BOXPLOT
graphical display of data using 5 number summary
(if outliers shown, called “modified box plot”)
OUTLIERS
Outliers:
IQR Test for Outliers:
(IQR )(1.5) = multiplier M
Q1 - M = outlier lower bound
Q3 + M = outlier upper bound
If values exceed these bounds, they are outliers
RESISTANCE
Resistant Measures:
Non-resistant Measures:
Mean, Midrange:
Median, IQR, Trimmed Mean:
MEASURES OF SPREAD
Range: the spread between high and low
Resistant?
IQR (Interquartile Range) :
Resistant?
STANDARD DEVIATION
a measure of the average amount of deviation from
the mean among the data values
Population St. Deviation:
Sample St. Deviation:
We generally use sx because we usually do not have
entire population.
VARIANCE
the square of the standard deviation
what you get before taking square root
Population Variance:
Sample Variance:
This measure not used much in elementary
statistics but you need to know what it is.
Coefficient of Variance
measure of how relatively large a st. dev. is
Ex: St. deviation of IQ = 15, Mean 100
St. deviation of height = 3 in, Mean 69
“Comment on the distribution”
You now have numbers to support your
statements, rather than just graphs.
SHAPE:
OUTLIERS:
CENTER:
SPREAD: how widely does the data vary?
Unusual Features: gaps, clusters
SHAPE
If the mean > median, then data distribution
is skewed ________The mean is in the tail.
If the mean < median, then data distribution
is skewed ________The mean is in the tail.
If the mean ≈ median, then data distribution
is approximately ____________.
SHAPE
Symmetric if mean = median
SKEWNESS
Skewed left if mean < median
Skewed right if mean > median
Left
Right
Mean is in the tail of the data
OTHER SHAPES
Uniform distribution: all values relatively
evenly distributed across interval
Bimodal distribution: two peaks
TRANSFORMATIONS TO DATA
What would happen to the statistical measures
if each data value had a constant added to or
subtracted from it?
Mean:
Standard Deviation:
Median:
IQR:
What would happen to the statistical measures
if each data value had a constant multiplied or
divided by it?
Mean:
Standard Deviation:
Median:
IQR:
TRANSFORMATIONS TO DATA SET
What would happen to the statistical measures
if one very low or very high data value was
added to the set?
Mean:
Standard Deviation:
Median:
IQR:
MEASURES OF POSITION
Give a numerical approximation of where a
single data value stands compared to the
whole distribution
Quartiles:
Percentiles:
Z Scores:
Z SCORES
standardized score
how a single value compares to entire data set
in terms of position in distribution
z=
How unusual are you?
Compute your z score for height?
Compute your z score for Math SAT?
Compute your z score for IQ?
NORMAL MODEL
shows how data is distributed symmetrically
along an interval according to empirical rule
Empirical Rule:
of data within 1 st. deviation of μ
of data within 2 st. deviations of μ
of data within 3 st. deviations of μ
ANOTHER OUTLIER TEST
Using Empirical Rule:
Data values of z > +2 st. deviations away
from mean are mild outliers
Data values of z > +3 st. deviations away
from mean are extreme outliers
NORMAL CURVE
a theoretical ideal about how traits/characteristics
are distributed
Many human traits are approximately normally
distributed such as height, body temp, IQ, pulse
Avoid using “normal” when describing data—say
“approximately normal or symmetric” unless
clearly mound-shaped, bell-shaped
NORMAL CURVE
Normal curve—symmetric, mound-shaped
Area under curve=
A z score can be used to establish what % of
the curve is less or more than the z score,
and establish probability of a data value being
in that position.
FINDING PERCENTILE/PROBABILITY
USING NORMAL CURVE
1. Calculate z score for data value
2. Use calculator: normalcdf under DISTR
key
Looking for area > z score: normalcdf (z, ∞)
Looking for area < z score: normalcdf (∞, z)
Looking for area between z scores:
normalcdf (z1, z2)
FINDING CUT OFF SCORES
If you are given a percentile or probability, and
need to determine the “cut off score”
1. Sketch curve to determine where z score is located.
2. Determine if you want area above or below this
percentile
3. Use INVNORM on calculator
invnorm(percentile)= z score
4. Use z score formula to solve for x.
Does the data fit a normal model?
1. Check mean and median
2. Make a NORMAL PROBABILITY PLOT—
3. Make a BOXPLOT on calculator.
AVOID using histograms on calculator to check.