standard deviation
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Transcript standard deviation
Statistical
analyses
SPSS
Statistical analysis program
It is an analytical software recognized by the scientific
world
(e.g.: the Microsoft Excel program is not recognized by
the scientific world)
SPSS
Let’s start the SPSS software!
Paste the data onto the DATA VIEW window!
It has two windows, one of them contains the data
(DATA VIEW), and the types of the variables must be
given in the other one (VARIABLES).
Exact coding of variables is the basis of successful SPSS
use.
Basics of
computer-based
analysis
Types of data
Measurable data
Differences between data are equal
E.g.
interval scale
How old are you?
How much is your weight?
Ordinal data
Data originating from gradation
Special type: reletad gradation positions
Nominal scale
The data are replaced by numbers.
E.g. Gender? 1. Male 2. Female
The data do not signal order
The data cannot be added
Statistical procedures
Descriptive statistics
If we analyze actual persons,
that is population = samples
Statistical indicators
Frequencies
Central tendency
Dispersion
Correlation
Statistical procedures
Mathematical statistics
It provides the information whether we may draw
conclusion based on the representative sample
referring to the population.
Definition
Population: the group which the conclusions
refer to
E.g.: university student; German people;
teachers
Sample: the ones actually involved in the
surveys
Representative sample: when the
composition of the sample mirrors the
composition of the population.
E.g.: Gallup’s deal with the Public Opinion
Office around the time of the presidential
elections in 1936
Mathematical statistics
Analysis of differences
The aims: to show the criteria in which
elements differ from each other
Types of data Scale
Ordinal
Nominal
Number of
samples
One
One-sample tsamples test
Wilcoxon-test
Crosstabs analysis,
Chi-square test
Two
Independent tsample
F-test
Mann-Whitney-test
Cross database analysis,
Chi-square test
Three or more
ANOVA analysis
Kruskall-Wallistest
Cross database analysis,
Chi-square test
Mathematical statistics
Analyzing correlations
Types of data Scale
Ordinal
Nominal
Spearman
correlate
Crosstabs analysis,
Chi-square test
Number of
samples
Two
Correlate
Two or more
Regression
More than two
Partial correlate
Factor analysis
Cluster analysis
Descriptive statistics
Central Tendency
Mean
Modus : (most frequent data)
Median
Frequency
1.
2.
Determining the number of categories
An odd number between 10 and 20
If the number of the samples is low (e.g.50 responders)
there can be fewer categories (7 categories)
Determining the intervals
1, 2, 3, 5, 10 depending on the number of categories
Disjunction: It should be noted that the each item in the
sample must be categorized into one particular category,
so the groups may not overlap.
E.g.: Bad samples:
Age groups
Below 20
20-30
30-40…
E.g.: God examples:
Age groups
Below 20
20-29
30-39…
Absolute frequency
Def: The number of items belonging to particular
category is absolute frequency value.
the subgroup frequencies together create the
absolute frequency distribution of the sample.
Further frequency indicators
Relative frequency means the quotient of the absolute
frequency values and the number of the samples.
The relative frequency gives the percent of the responders
in one particular category compared to the total number of
samples.
Cumulative frequency means how many items of the
sample can be found all together below the upper limit of
the category.
Cumulative percent means the quotient of the cumulative
frequency and the number of the sample.
IT shows what percent of the sample can be found below
the upper limit of the category.
Dispersion indicators
Range: the range of the samples means the
difference between the highest and lowest
items.
R = Xmax - Xmin
Average difference:
the average distance (absolute deviation) of the
items from the average.
Square sum:
Sum of the quadrant of the deviation from the
average.
Variance
Variance
the square sum divided by the degree of
freedom of the sample
Degree of freedom is the number of the
independent elements (the number of the
responders) of the sample.
Standard deviation
Standard deviation is the square root
with a positive sign of the variance.
Theorem
More than 2/3 of the data belong to a 1
standard deviation extending to the positive and
negative directions from the mean.
More than 90% of the data belong to a 2
standard deviation taken from the mean.
More than 90% of the data belong to a 3
standard deviation taken from the mean.
Relative standard deviation
The Relative deviation is an indicator
related which provides what percent of
the mean is the standard deviation.
Relative deviation =
standard deviation
mean
Quartiles
The quartiles are the quartering points of
the sample.
Interquartiles half-extension: is the
difference between the third and the first
quartile: Q3-Q1
Interrelations
Interrelations between
frequency and mean indicator
Left tendency: Modus > Median > Mean
Right tendency : Modus < Median < Mean
Interrelations between
frequency and mean indicatior
Normal distribution (bell curve) :
All the three indicators coincide
Modus = Median = Mean
Mathematical statistics
Relations examinations
Correlation
Correlation coefficient is the indicator which
shows the direction and strength between two
data list.
Correlation
rxy rtáblázat
There is correlation between the two samples
rxy rtáblázat
There is no correlation between the two samples
Correlation coefficient
The interpretation of the correlation coefficient
0,9 – 1
0,75 – 0,9
0,5 – 0,75
0,25 – 0,5
0,0 – 0,25
extremely strong correlation between the
two data lists
strong
detectable
weak
no relationship
Direction
If the correlation coefficient is negative contrasting relationship
E.g. The numbers of hours doing sports – your weight
If the correlation coefficient is positive data changing
simultaneously
28
E.g. The size of your home library – the rate of loving to read
Relationship between/among
variables –
Crosstabs
Crosstabs – illustrating the distribution of two
nominal or ordinal variables on the same
chart.
Crosstabs- Chi-square
It is an indicator which shows whether the
correlations in the cross tabs are valid only for the
samples or for the population as well.
It cannot be used efficiently if the value is less then
5 in more than 20% of the cells.
Hypothesis analyses
It is a method to decide whether the
differences in data are significant or random.
Paired-samples T-test
The paired-samples T-test is used when the same people are
asked or tested twice (e.g. one-sample experiment)
z
t n
s
'
Where:
z
- mean
s - Standard deviation
Paired-samples T-test
Match the t-number with the value of the
„Critical values of the t-distribution” chart
If t’ > t chart the different is significant
If t’ < t chart the different is random
T-test with computer
It is not necessary use the „Critical values of
the t-distribution” chart, because most
software provides the „p” value (Signif of t,
Sig.Level).
The „p” shows what percent is the failure
rate.
If „p”<0.05 (5%) then the difference is
significant
Independent t-test
H0: two independent samples taken from the
same population.
(H0 definition: the zero hypothesis is that
the difference is random )
This type of test can only can be conductived
if the variances of the two groups not too
different.
The F-test can give the answer.
F-test
s12
F 2
s2
The F-test is the quotient of the variance squares.
If Fnumber < Fchart there is no significant
difference
If Fnumber > Fchart there is a great difference
between the variances
the T-test cannot be done.
you can try the Welch-test.
Independent t-test
t
xy
n
m
2
(
x
x
)
(
y
y
i
)
i
2
i 1
i 1
nm2
The degree of freedom = n+m-2.
nm
nm
Illustration of result
Egyéni eredmény
Histogram
9
12
15
6
18
24
Missing
5
3
REL
2
1
6
Mean = 12,9
Std. Dev. = 5,515
N = 20
0
0
5
10
15
20
25
5
30
REL
4
Frequency
Frequency
4
3
2
1
Mean = 12,9
Std. Dev. = 5,515
N = 20
0
0
5
10
15
20
REL
Aim: to make the result look
conceivable and visual
25
30
Frequency polygon
Illustrating frequency data with a line diagram.
Histogram
Illustrating frequency data with a bar diagram.
The title of the X axis is intervals.
Histogram shapes
Symmetrical, normal
Symmetrical, peaked
Histogram shapes
bimodal
Histogram shapes
Right side tendency
Histogram shapes
Left side tendency
Interrelations between frequency and
mean indicator
Normal distribution: Mean = Median = Modus
Skewness = 0
Interrelations between frequency and mean
indicator
Symmetric with two modes
Bimodul
Skewness = 0
Interrelations between frequency and mean
indicator
Right side tendency
Mode<Median<Mean
Skewness = (-)
Interrelations between frequency and mean
indicator
Right side tendency
Mean < Median < Mode
Skewness = (+)
Normal distribution with different
standard deviation
Kurtosis = 1 normal
If the Kurtosis value is bigger the
polygon is flatter