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%.

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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.

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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

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μ – mean, medianσ2 – varianceσ – standard deviation

**The red line is the standard normal distribution

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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