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
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Mean: arithmetic average of all data values
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
5 important numbers in data set:
Q1, Med, Q3, may not be actual data values
graphical display of data using 5 number summary
(if outliers shown, called “modified box plot”)
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
Median, IQR, Trimmed Mean:
Range: the spread between high and low
IQR (Interquartile Range) :
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.
the square of the standard deviation
what you get before taking square root
This measure not used much in elementary statistics but you need to know what it is.
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
You now have numbers to support your statements, rather than just graphs.
SPREAD: how widely does the data vary?
Unusual Features: gaps, clusters
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 ____________.
Symmetric if mean = median
Skewed left if mean < median
Skewed right if mean > median
Mean is in the tail of the data
Uniform distribution: allvalues relatively
evenly distributed across interval
Bimodal distribution: two peaks
What would happen to the statistical measures if each data value had a constant added to or subtracted from it?
What would happen to the statistical measures if each data value had a constant multiplied or divided by it?
What would happen to the statistical measures if one very low or very high data value was added to the set?
Give a numerical approximation of where a single data value stands compared to the whole distribution
how a single value compares to entire data set
in terms of position in distribution
Compute your z score for height?
Compute your z score for Math SAT?
Compute your z score for IQ?
shows how data is distributed symmetrically along an interval according to 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
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
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—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.
Looking for area > z score: normalcdf (z, ∞)
Looking for area < z score: normalcdf (∞, z)
Looking for area between z scores:
normalcdf (z1, z2)
If you are given a percentile or probability, and
need to determine the “cut off score”
2. Determine if you want area above or below this percentile
3. Use INVNORM on calculator
invnorm(percentile)= z score
2. Make a NORMAL PROBABILITY PLOT—
3. Make a BOXPLOT on calculator.
AVOID using histograms on calculator to check.