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brittanyball42
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Raw scores
they are meaningless because we dont know whats good or bad, high or low
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Relative scale
- most test scores are judged on a relative scale
- relative to other test takers
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Summary statistics
- about summarizing single variables
- focus on quantitative (numerical)variables
- start with a "bag of data" (collection of numbers)
- consists of one or (usually) more variables
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what does summarizing mean
- making information more concise (shorter)
- summarizing depends on the sample size (N)
- if N is large, we need to be very concise
- if N is small, we can be less concise (more complete)
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Sorting
- the simpliest summary technique is to sort the data
- works with small sets of numbers
- easier to see the distribution when the data are sorted
- no information is lost; the presentation is merely simplified
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Histogram
- a bar graph of a grouped frequency distribution of quantitative variable
- the apperance of a histogram can vary depending on how many categories you use
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how to create a histogram
- create categories or groups of bins
- count the number of people or items in each group
- make a bar graph, one bar for each group
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Frequency polygons
- the same as histograms, but midpoints connected by lines, rather than using bars
- not used very much
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raw frequencies
- counts
- are the original numbers
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relative frequencies
- the numbers divided by N (the total)
- percentages are the same relative frequency, except with the decimal point shifted over two places
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Symmetrical
- left side is the mirror image of the right side
- many distributions are symmetrical
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Shapes of distributions
- Symmetrical
- Uniform
- Bell-shaped
- Floor and Ceiling effects
- Skewed
- Bimodal
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Uniform
- equal probabilities in all categories
- uniform distribution is symmetrical
- bars are close together in uniform
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Bell-shaped
- most common
- another examole of a symmetrical distribution
- bars are close togther in a bell shape
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Floor effects
- there is a lower limit to the possible numbers
- usually this is 0
- examples: incomes, which generally cannot be negative
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ceiling effect
an upper limit to the possible numbers
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Skewed
- to the right (positively skewed)
- to the left (negatively skewed)
- skew us frequentky due to floor and ceiling effects
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Bimodal
- two humps or central points
- like two bell shaped put together
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Boxplots (or box-and-whisker plots)
- includes median (a small square)
- outliers (small circle)
- non-outlier range (in the shape of a capital I)
- and the percentage (a big box)
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measures of central tendency
- these measure where the "middle" or "center" is, or where most of the action is in the distribution
- includes the mean, median, and mode
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measures of dispersion or variability
theses measure how spread out the data are
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mean
- arithmetic average- add them up and divide by N
- most sensitive to outliers
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median
- middle-most number (same as the 50th percentile)
- if there is an even amount of numbers, average the middle two
- sort the numbers first
- less sensitive to outliers
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mode
- the most frequently occuring number.
- the hump in the histograms
- the only measure that works with qualitative data
- the only measure of central tendency where there can be two (eg. bimodal)
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when a distribution is symmetrical and bell-shaped
the mean median and mode are the same
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when distributions are skewed
mean, median, and mode are separate
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measures of dispersion of variability
- these measure how spread out the data are
- a data set: 3 3 3 3 (0 variability)
- another data set: 1 2 3 4 5 (medium variability)
- another data set: -1 1 3 5 7 (larger variability)
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Ordinal measures of variability
- these depend only on the order of the numbers
- range, interquartile range, and semi-interquartile range
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interquartile range
- chop off the top 25% (upper quartile)
- chop off the bottom 25% (lower/bottom quartile)
- take the difference
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semi-interquartile range
half of the interquartile range
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quantitative measures of variability
- these are based on the actual numbers, not just their orders
- variance and standard deviation
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variance
average squared deviation from the mean
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Standard deviation
square root of the variance
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Norms
- are summary statistics of test results-they tell us what is "normal" or average
- we can tell how far an individual score is from average using summary statistics
- Z scores are commonly used
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