"We all marvel at the beauty of the Butterfly, but rarely take into account the changes it has undergone to get there."

- Axotlyorill

Originally by "‘piccolojunior"’ on the College Conﬁdential forums; reformatted/reorganized/etc by Dillon Cower. Comments/suggestions/corrections: dcower@gmail.com

1

¯ • Mean = x (sample mean) = µ (population mean) = sum of all elements ( number of elements (n) in a set = of center.

x n

x) divided by

. The mean is used for quantitative data. It is a measure

• Median: Also a measure of center; better ﬁts skewed data. To calculate, sort the data points and choose the middle value. • Variance: For each value (x) in a set of data, take the difference between it and the mean (x − µ ¯ or x − x ), square that difference, and repeat for each value. Divide the ﬁnal result by n (number of elements) if you want the population variance (σ2 ), or divide by n − 1 for sample variance (s2 ). Thus: Population variance = σ2 =

(x−µ)2 . n (x−µ)2 . n

Sample variance = s2 =

(x−¯ )2 x . n−1

• Standard deviation, a measure of spread, is the square root of the variance. Population standard deviation = σ2 = σ = Sample standard deviation = s2 = s =

(x−¯ )2 x . n−1 σ . n

– You can convert a population standard deviation to a sample one like so: s = • Dotplots, stemplots: Good for small sets of data. • Histograms: Good for larger sets and for categorical data. • Shape of a distribution:

– Skewed: If a distribution is skewed-left, it has fewer values to the left, and thus appears to tail off to the left; the opposite for a skewed-right distribution. If skewed right, median < mean. If skewed left, median > mean. – Symmetric: The distribution appears to be symmetrical. – Uniform: Looks like a ﬂat line or perfect rectangle. – Bell-shaped: A type of symmetry representing a normal curve. Note: No data is perfectly normal - instead, say that the distribution is approximately normal.

2

• Z-score = standard score = normal score = z = number of standard deviations past the mean; used for normal distributions. A negative z-score means that it is below the mean, whereas a...

1

¯ • Mean = x (sample mean) = µ (population mean) = sum of all elements ( number of elements (n) in a set = of center.

x n

x) divided by

. The mean is used for quantitative data. It is a measure

• Median: Also a measure of center; better ﬁts skewed data. To calculate, sort the data points and choose the middle value. • Variance: For each value (x) in a set of data, take the difference between it and the mean (x − µ ¯ or x − x ), square that difference, and repeat for each value. Divide the ﬁnal result by n (number of elements) if you want the population variance (σ2 ), or divide by n − 1 for sample variance (s2 ). Thus: Population variance = σ2 =

(x−µ)2 . n (x−µ)2 . n

Sample variance = s2 =

(x−¯ )2 x . n−1

• Standard deviation, a measure of spread, is the square root of the variance. Population standard deviation = σ2 = σ = Sample standard deviation = s2 = s =

(x−¯ )2 x . n−1 σ . n

– You can convert a population standard deviation to a sample one like so: s = • Dotplots, stemplots: Good for small sets of data. • Histograms: Good for larger sets and for categorical data. • Shape of a distribution:

– Skewed: If a distribution is skewed-left, it has fewer values to the left, and thus appears to tail off to the left; the opposite for a skewed-right distribution. If skewed right, median < mean. If skewed left, median > mean. – Symmetric: The distribution appears to be symmetrical. – Uniform: Looks like a ﬂat line or perfect rectangle. – Bell-shaped: A type of symmetry representing a normal curve. Note: No data is perfectly normal - instead, say that the distribution is approximately normal.

2

• Z-score = standard score = normal score = z = number of standard deviations past the mean; used for normal distributions. A negative z-score means that it is below the mean, whereas a...

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