Sample Standard Deviation
preferred measure of variation when the mean is used as the measure of center
takes into account all the observations
first step is find the deviations of the mean
Measures of center
value at the center or middle of the data set
mean
adding all of the values and dividing the total by the number of values
mode
most frequently occurring value
no modes if no repeating numbers
n denotes
number of data values in a sample
Median
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x bar
mean of a set of sample values
Range
difference between maximum and minimum
most frequently used measures of variation
range and sample standard deviation
standard deviation formula
Sample Variance formula
relationship between variance and standard deviation
the more variation that there is in a data set, the larger is its standard deviation
Definition of descriptive measures
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Procedures in computing a sample standard deviation
1. Calculate sample mean (x bar)
2. Find the deviation from the mean of each observations: xi - x bar
3. Sum of squared deviations
4. Sample variance
5. Sample standard deviation
(just take the sample variance and then take the square root of it)
Two-Standard-Deviations Rule
almost all the observation in any data set lit within two standard deviations to either side of the mean
five-number summary
bimodal
two data values with the same greatest frequency
multimodal
more than two data values occur with the same greatest frequency
no mode
no data is repeated
midrange
value midway b/w max & min value
MR=max+min/2
round off rule for measures of center
carry one more decimal place than is present in original set of values
mean from frequency distribution equation
x bar= E(f*x)/Ef
Weighted mean
when data values are assigned different weights, w, can compute a weighted mean
weighted mean equation
x bar= E(w*x)/Ew
round off rule
carry one more decimal place than is present (only for final answer)
Standard deviation
set of sample values, denoted by "s", measure of how much data values deviate away from the mean
shortcut formula SD
s=square root n(Ex^2)-(Ex)^2/
n(n-1)
range rule of thumb
the majority (95%) of sample values lies within two standard deviations of the mean
minimum (usual) value
0
Maximum (unusual) value
0
estimation range rule of thumb
s=range/4
population standard deviation equation
o= square root E(x-u)/N
variance
set of values is a measure of variation equal to the square of the SD
s
sample standard deviation
s^2
sample variance
o
Population standard deviation
o^2
Population variance
Empirical rule
-68% within 1 SD of the mean
-95% within 2 SD of the mean
-99.7% with 3 SD of the mean
Chebyshev's Theorem
Proportion of any data set lying with K SD of the mean is always at least 1-1/^2, where k is any positive number <1
Coefficient of Variation (CV)
for a set of nonnegative sample or population data, expresses as a percentage, is the mean
CV=s/x *100 <sample
CV=o/u*100 <population
z-score
standardized value, number of standard deviations that given value x is above or below the mean
measures of z-score
sample: z=x-x bar/s
Population: z=x-u/o
interpreting z-scores
unusual -3
ordinary=0
unusual=3
percentiles
measures of location p1,p2...p99 which divides a set of data into 100 groups with about 1% of the values in each group
finding percentile
percentile of value x= # of values <x/ total # of values times 100
converting from the k percentile
L=k/100 times n
percentile notations
n= total # of values in a data set
k= percentile being used
L= locater gives you position of value
Pk= kth percentile
quartiles
measures of located, denoted Q1,Q2,Q3, which divide a set of data into four groups with about 25% of the values in each group
Q1
first quartile, separates the bottom 25% of sorted values from the top 75%
Q2
second quartile, same as medium , separates at bottom 50% and top 50%
Q3
third quartile, separates bottom 75% from top 25%
interquartile(IQR)
Q3-Q1
semi-interquartile range
Q3-Q1/2
mid quartile
Q3+Q1/2
outliers
a value that lies very far away from the majority of the other values