Download ppt

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.

2
(x  )


n
2
μ – mean
x – data points
n – 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.
1 n
2

( xi   )

n i 1
• 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.
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
   ( x) 

1
 

e
( x )
 
1 x
 
, x  R
   
The continuous probability density function of the normal
distribution is the Gaussian function with μ = 0 and σ = 1
 0,1 ( x) 
e
 x2 / 2
2
Marina Maksimova
,xR
μ – mean, median
σ2 – variance
σ – standard deviation
**The red line is the standard normal distribution
Marina Maksimova
μ – mean, median
σ2 – variance
σ – standard deviation
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
Marina Maksimova