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| numbers that are used to describe data sets |
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| measure of centeral tendency (measures of center) |
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descriptive measures that indicates where the center, or most typical value, in a data set lie
often called averages |
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| sum of the observations divided by the number of observations |
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arrange the data in increasing order.
If the number of observations is odd, then the median is the observation exactly in the middle of the ordered list.
If the number of observations is even, then the median is the mean of the two middle observations in the ordered list
In both cases, if we let n denote the number of observations, then the median is at position (n+1)/2 in the ordered list |
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find the frequency of each value in the data set
if no value occurs more than once, then the data set has no mode
otherwise, any value that occurs with the greatest frequency is a mode of the data set |
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| not sensitive to the influence of a few extreme observations (e.g. median, but not mean) |
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more resistant mean
created by removing a percentage of the smallest and largest observations before computing the mean |
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for a variable x, the mean of the observations for a sample is called a sample mean and is denoted as an x with a line over it
mean of the sample data |
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| measures of variation (measures of spread) |
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| descriptive measures that indicate the amount of variation or spread in a data set |
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| difference between the maximum (largest) and minimum (smallest) observations |
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| measures variation by indicating how far, on average, the observations are from the mean |
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| how far each observation is from the mean |
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| sum of squared deviations |
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Definition
the sum of the squared deviations from the mean
gives a measure of the total deviation from the mean for all the observations |
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| valid for all data sets and implies, in particular, that at least 89% of the observations lie within three standard deviations to either side of the mean |
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| If the distribution of the data set is approximately bell shaped, then we can apply this rule, which implies, in particular, that roughly 99.7% of the observations lie within 3 standard deviations to either side of the mean |
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| divide a data set into hundredths, or 100 equal parts |
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| divide a data set into tenths, or 10 equal parts |
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| divide a data set into fifths, or 5 equal parts |
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| divide a data set into quarters, or 4 equal parts |
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| the number that divides the bottom 25% from the top 75% |
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the number that divides the bottom 50% from the top 50%
median |
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Definition
| number that divides the bottom 75% from the top 25% |
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| interquartile range (IQR) |
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Definition
| the difference between the first and third quartiles (Q3 - Q1) |
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| observations that fall well outside the overall pattern of data |
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| observations that fall below the lower limit or above the upper limit |
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| boxplot (box-and-whisker diagram) |
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| based on the five-number summary and can be used to provide a graphical display of the center and variation of a data set |
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| constructing a boxplot procedure |
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Definition
1. determine the quartiles
2. determine potential outliers and the adjacent values
3. draw a horizontal axis on which the numbers obtained in steps 1 and 2 can be located. above this axis, mark the quartiles and the adjacent values with vertical lines.
4. connect the quartiles to make a box, and then connect the ox to the adjacent values with lines
5. plot each potential outlier with an asterisk |
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Definition
| two lines emanating from the box in a boxplot |
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| population mean (mean of a variable) |
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Definition
| for a variable, x, the mean of all possible observations for the entire population |
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| population standard deviation (standard deviation of a variable) |
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Definition
| for a variable, x, the standard deviation of all possible observations for the entire population |
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| a descriptive measure for a population |
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| a descriptive measure for a sample |
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always has a mean of 0 and standard deviation of 1
the standardized version of a variable x is obtained by first subtracting from x its mean and then dividing by its standard deviation |
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Definition
for an observed value of a variable, x, the corresponding value of the standarized variable z is called the z-score of the observation. The term standard score is often used instead of z-score.
A negative z-score indicates that the observation is below the mean, whereas a positive score indicates that the observation is above the mean |
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