Marina Maksimova Session 7 Comments for “Classical Tests”
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Transcript of Marina Maksimova Session 7 Comments for “Classical Tests”
Marina Maksimova
Session 7
Comments for
“Classical Tests”
Marina Maksimova
Mean, Median, Mode –- are three kinds of "averages".
• The mean is the sum of the observations divided by the number of observations.
• The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one.
• The mode of a data sample is the element that occurs most often in the collection.
Marina Maksimova
Variance
• The variance and the closely-related standard deviation are measures of how spread out a distribution is. In other words, they are measures of variability.
• The variance is computed as the average squared deviation of each number from its mean.
n
x
22
)( μ – mean
x – data pointsn – number of data points in set
Marina Maksimova
Standard deviation
• The standard deviation formula is very simple: it is the square root of the variance. It is the most commonly
used measure of spread.
• In a normal distribution, about 68% of the scores are within one standard deviation of the mean and about 95% of the scores are within two standard deviations of the mean.
n
iixn 1
2)(1
Marina Maksimova
Standard deviation (cont’d)
• Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for about 68% of the set (dark blue), while two standard deviations from the mean (medium and dark blue) account for about 95%, and three standard deviations (light, medium, and dark blue) account for about 99.7%.
Marina Maksimova
Normal Distribution
• In probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that describes data that clusters around a mean or average.
• The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known as the Gaussian function or bell curve.
Marina Maksimova
Probability density function
Rxx
exx
,
11)(
)(
Rxe
xx
,2
)(2/
1,0
2
The continuous probability density function of the normal distribution is the Gaussian function with μ = 0 and σ = 1
μ – mean, medianσ2 – varianceσ – standard deviation
Marina Maksimova
μ – mean, medianσ2 – varianceσ – standard deviation
**The red line is the standard normal distribution
Marina Maksimova
P-value
• A p-value is an estimate of the probability that a particular result, or a result more extreme than the result observed, could have occurred by chance, if the null hypothesis were true.
• In short, the p-value is a measure of the credibility of the null hypothesis.
• If something is sufficiently unlikely to have occurred by chance (say, p<0.05),we say that it is statistically significant.
Marina Maksimova
Degrees of freedom
• In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.
• DF = n – k,
• n - sample size• k - number of parameters, estimated from the
data
Marina Maksimova
Reference
• Wikepedia.com