2. Descriptive Statistics
Measures of Central Tendency
Mean--------Interval or Ratio scale Polygon
– The sum of the values divided by the number of
values--often called the "average." μ=ΣX/N
– Add all of the values together. Divide by the number
of values to obtain the mean.
– Example:
X
7
12
24
20
19
????
2
4. The Characteristics of Mean
1. Changing a score in a distribution will
change the mean
2. Introducing or removing a score from
the distribution will change the mean
3. Adding or subtracting a constant from
each score will change the mean
4. Multiplying or dividing each score by a
constant will change the mean
5. Adding a score which is same as the
mean will not change the mean
4
5. Descriptive Statistics
Measures of Central Tendency
Median/MiddleOrdinal ScaleBar/Histogram
– Divides the values into two equal halves, with
half of the values being lower than the median
and half higher than the median.
Sort the values into ascending order.
If you have an odd number of values, the
median is the middle value.
If you have an even number of values, the
median is the arithmetic mean (see above) of
the two middle values.
– Example: The median of the same five numbers
(7, 12, 24, 20, 19) is ???.
5
6. Statistics
The median is 19.
Mode-Nominal Scale Bar/Histogram
–The most frequently-occurring value (or
values).
Calculate the frequencies for all of the
values in the data.
The mode is the value (or values) with
the highest frequency.
–Example: For individuals having the
following ages -- 18, 18, 19, 20, 20, 20, 21,
and 23, the mode is ???? The Mode is 20
6
8. The Range
The Range:
The Range is the difference between
the highest number –lowest number +1
2, 4, 7, 8, and 10 -> Discrete Numbers
2, 4.6, 7.3, 8.4, and 10 -> Continues
Numbers
The difference between the upper real
limit of the highest number and the
lower real limit of the lowest number.
11. Variability
Variability is a measure of
dispersion or spreading of
scores around the mean, and
has 2 purposes:
1. Describes the distribution
Next slide
11
12. Range, Interquartile Range, Semi-Interquartile
Range, Standard Deviation, and Variance are the
Measures of Variability
The Range:
The Range is the difference between the
highest number –lowest number +1
2, 4, 7, 8, and 10 -> Discrete Numbers
2, 4.6, 7.3, 8.4, and 10 -> Continues
Numbers
The difference between the upper real
limit of the highest number and the lower
real limit of the lowest number.
13. Variability
2. How well an individual score (or
group of scores) represents the
entire distribution. i.e. Z Score
Ex. In inferential statistics we
collect information from a small
sample then, generalize the results
obtained from the sample to the
entire population.
13
14. Interquartile Range (IQR)
In descriptive statistics, the
Interquartile Range (IQR),
also called the midspread or
middle fifty, is a measure of
statistical dispersion, being
equal to the difference
between the upper and lower
quartiles. (Q3 − Q1)=IQR 14
18. Interquartile Range (IQR)
IQR is the range covered
by the middle 50% of the
distribution.
IQR is the distance
between the 3rd Quartile
and 1st Quartile.
18
22. Variability
Range, SS, Standard Deviations and Variances
X σ² = ss/N Pop
1 σ = √ss/N
2
4 s² = ss/n-1 or ss/df Standard deviation
5 s = √ss/df Sample
SS=Σx²-(Σx)²/N Computation
SS=Σ( x-μ)² Definition
Sum of Squared Deviation from Mean
Variance (σ²) is the Mean of Squared Deviations=MS22
23. 23
Practical Implication for Test
Construction
Variance and Covariance measure the quality of each
item in a test.
Reliability and validity measure the quality of the
entire test.
σ²=SS/N used for one set of data
Variance is the degree of variability
of scores from mean.
Correlation is based on a statistic called Covariance (Cov xy
or S xy) ….. r=sp/√ssx.ssy
COVxy=SP/N-1 used for 2 sets of data
Covariance is a number that reflects the degree to
which 2 variables vary together.
24. 24
Variance
X σ² = ss/N Pop
1 s² = ss/n-1 or ss/df Sample
2
4
5
SS=Σx²-(Σx)²/N
SS=Σ( x-μ)²
Sum of Squared Deviation from Mean
25. 25
Covariance
Correlation is based on a statistic called
Covariance (Covxy or Sxy) …..
COVxy=SP/N-1
Correlation-- r=sp/√ssx.ssy
Covariance is a number that reflects the
degree to which 2 variables vary
together.
Original Data
X Y
1 3
2 6
4 4
5 7
26. 26
Covariance
Correlation is based on a statistic called
Covariance (Covxy or Sxy) …..
COVxy=SP/N-1
Correlation-- r=sp/√ssx.ssy
Covariance is a number that reflects the
degree to which 2 variables vary
together.
Original Data
X Y
8 1
1 0
3 6
0 1
31. FACTORS THAT AFFECT
VARIABILITY
1. Extreme Scores i.e. 1, 3, 8, 11, 1,000,000.00 . We can’t use
the Range in this situation but we can use the other measures of
variability.
2. Sample Size If we increase the sample size will change the
Range therefore we can’t use the Range in this situation but we can
use the other measures of variability.
3. Stability Under Sampling (see next slide) p.130 The
S and S² for all samples should be the same because they come from
same population (all slices of a pizza should taste the same).
4. Open-Ended Distribution When we don’t have
highest score and lowest score in a distribution
31