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35 Cards in this Set

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Central tendency

a single value used to describe the center point of a data set
Measures of central tendency
mean, median, and mode
Mean
AKA average, is the most common measure of central tendency;
add all values in a data set and divide the result by the number of observations
x- bar = sample mean
xi = the values in the sample
n = the number of data values in the sample
Mu - the population mean
N - the number of data values in the population
Weighted mean
allows you to assign more weight to certain values and less weight to others
wi - the weight for each data value xi
bottom of formula - the sum of all the weights
Advantages of using mean to summarize data
simple to calculate
summarizes the data with a single value
disadvantages of using the mean to summarize data
with only a summary value you lose information about the original data;
the value of the mean is sensitive to outliers
median
the value in the data set for which half the observations are lower and half are higher;
NOT sensitive to outliers
formula for the index point for the median
i = .5(n)
whenever the index point isn't a whole number, round the value up to the next highest whole number
mode
value that appears most often in a data set;
if no data value or category repeats, we say the mode doesn't exist;
more than one mode can exist if 2 or more values tie for most frequent;
particularly useful way to describe categorical data
Which measure of central tendency should you use?
mean is generally used, unless extreme outliers exist;
if outliers are present, the median is often used, since the median isn't sensitive to outliers;
for categorical data, the mode is the best choice
Measures of variability
show how much spread is present in the data;
range, variance, and standard deviation
range
simplest measure of variation;
difference between the highest value and the lowest value in a data set
s^2 =
sample variance
Sample variance formula
(xi - xbar) = the difference between each data value and the sample mean
(xi - xbar) = the difference between each data value and the sample mean
Standard deviation
the square root of the variance;
has the same units at the original data
the square root of the variance;
has the same units at the original data
Sample standard deviation formula
Population variance formula
coefficient of variation (CV)
measures the standard deviation in terms of its percentage of the mean;
a high cv indicates high variability relative to the size of the mean and vice-versa;
a smaller coefficient of variation indicates more consistency within a set of data values
sample coefficient of variation formula
population coefficient of variation formula
z-score
identifies the number of standard deviations a particular value is from the mean of its distribution;
has no units;
0 for values equal to the mean;
positive for values above the mean;
negative for values below the mean;
z-score of an outlier
is above +3 or below -3
population z-score formula
sample z-score formula
The empirical rule
if a distribution follows a bell-shaped, symmetrical curve centered around the mean, we would expect:
approx 68% of the values to fall within +/- 1 standard deviations from the mean;
approx 95% of the values to fall within +/- 2 standard deviations from the mean
approx 99.7% of the values to fall within +/- 3 standard deviations from the mean
Measures of relative position
Measures of relative position compare the position of one value in relation to other values in the data set;
Percentiles and quartiles
Percentiles
measure the approx percentage of values in the data set that are below the value of interest
find percentiles manually
sort the data from lowest to highest;
calculate the index point, i;
if i is not a whole number, round it to the next whole number;
if i is a whole number, the midpoint between i and i+1 position is our value
percentile rank
identifies the percentile of a particular value within a set of data
percentile rank formula
quartiles
split the ranked data into 4 equal groups:
the 1st quartile (Q1) constitutes the 25th percentile;
the 2nd quartile (Q2) constitutes the 50th percentile (also is the median);
the 3rd quartile (Q3) constitutes the 75th percentile
Interquartile Range (IQR)
describes the middle 50% of a range;
find the IQR by subtracting the first quartile from the third quartile;