Data analysis

1
Data analysis
2

• The first step in any data analysis strategy is
  to calculate summary measures to get a general
  feel for the data. Summary measures for a data
  set are often referred to as descriptive
  statistics. Descriptive statistics fall into
  three main categories
• measures of position (or central tendency)
• measures of variability
• measures of skewness

3

• The purpose of descriptive statistics is to
  describe the data.
• The type of data will determine which descriptive
  statistic is appropriate.
• Specifically, one can only calculate a mean with
  interval or ratio data, whereas a mode can be
  calculated with nominal, ordinal, interval or
  ratio data.

4
Measures of Position

• Measures of position (or central tendency)
  describe where the data are concentrated.
• MeanThe Mean is simply the mathematical average
  of the data. T
• the mean provides you with a quick way of
  describing your data, and is probably the most
  used measure of central tendency.
• However, the mean is greatly influenced by
  outliers. For example, consider the following
  set 1 1 2 4 5 5 6 6 7 150
• While the mean for this data set is 18.7, it is
  obvious that nine out of ten of the observation
  lie below the mean because of the large final
  observation.
• Consequently, the mean is not always the best
  measure of central tendency.

5

• MedianThe median is the middle observation in a
  data set. That is, 50 of the observation are
  above the median and 50 are below the median
  (for sets with an even number of observation, the
  median is the average of the middle two
  observation).
• The median is often used when a data set is not
  symmetrical, or when there are outlying
  observation.
• For example, median income is generally reported
  rather than mean income because of the outlying
  observation.

6

•  
• To get the median, first put your numbers in
  ascending or descending order. Then just use
  check to see which of the following two rules
  applies
• Rule One. If you have an odd number of numbers,
  the median is the center number (e.g., three is
  the median for the numbers 1, 1, 3, 4, 9).  
• Rule Two. If you have an even number of numbers,
  the median is the average of the two innermost
  numbers (e.g., 2.5 is the median for the numbers
  1, 2, 3, 7).  

7

• ModeThe Mode is the value around which the
  greatest number of observation are concentrated,
  or quite simply the most common observation.
• Mode is often used with nominal data, but is not
  the preferred measure for other types of data.

8

• The mean, median, and mode are affected
  differently by skewness (i.e., lack of symmetry)
  in the data.

9

• When a variable is normally distributed, the
  mean, median, and mode are the same number.  

10

• When the variable is skewed to the left (i.e.,
  negatively skewed), the mean is pulled to the
  leftthe most, the median is pulled to the left
  the second most, and the mode the least affected.
• Therefore, mean lt median lt mode.

11

• When the variable is skewed to the right (i.e.,
  positively skewed), the mean is pulled to the
  right the most, the median is pulled to the right
  the second most, and the mode the least affected.
  
• Therefore, mean gt median gt mode.

12
Measures of Variability

• While measures of position describe where the
  data points are concentrated, measures of
  variability measure the dispersion (or spread) of
  the data set.
• Range
• The range is the difference between the largest
  and the smallest observations in the data set.
  However, This is a limited measure because it
  depends on only two of the numbers in the data
  set. Using the above data set again, the range is
  149, but that does not provide any information
  regarding the concentration of the data at the
  low end of the scale. Another limitation of range
  is that it is affected by the number of
  observations in the data set.
• Generally, the more observation there are, the
  more spread out they will be. One use of range in
  everyday life is in newspaper stock market
  summaries, which give the day's high and low
  numbers.

13

• Measures of Variability
• Measures of variability tell you how "spread out"
  or how much variability is present in a set of
  numbers.
• For example, which set of the following numbers
  appears to be the most spread out?
• Set A.  93, 96, 98, 99, 99, 99, 100
• Set B.  10, 29, 52, 69, 87, 92, 100
• Right! The numbers in set B are more "spread
  out."
• One crude indicator of variability is the range
  (i.e., the difference between the highest and
  lowest numbers).

14

• Two commonly used indicators of variability are
  the variance and the standard deviation.
• VarianceUnlike range, variance takes into
  consideration all the data points in the data
  set. If all the observation are the same, the
  variance would be zero. The more spread out the
  observation are, the larger the variance.
• The variance tells you (exactly) the average
  deviation from the mean, in "squared units."

15

• Standard DeviationStandard deviation is the
  positive square root of the variance, and is the
  most common measure of variability. Standard
  deviation indicates how close to or how far the
  numbers tend to vary from the mean. The larger
  the standard deviation, the more variation there
  is in the data set.
•  
• (If the standard deviation is 7, then the
  numbers tend to be about 7 units from the mean.
  If the standard deviation is 1500, then the
  numbers tend to be about 1500 units from the
  mean.)

16

• Virtually everyone in education is already
  familiar with the normal curve
• An easy rule applying to data that follow the
  normal curve is the "68, 95, 99.7 percent rule."
  That is . . .  
•  Approximately 68 of the cases will fall within
  one standard deviation of the mean.
• Approximately 95 of the cases will fall within
  two standard deviations of the mean.
• Approximately 99.7 of the cases will fall within
  three standard deviations of the mean.

17

• Higher values for both of these indicators stand
  for a larger amount of variability. Zero stands
  for no variability at all (e.g., for the data 3,
  3, 3, 3, 3, 3, the variance and standard
  deviation will equal zero).

18

• Frequency Distributions
• One useful way to view information in a variable
  is to construct a frequency distribution (i.e.,
  an arrangement in which the frequencies, and
  sometimes percentages, of the occurrence of each
  unique data value are shown).
• When a variable has a wide range of values, you
  may prefer using a grouped frequency distribution
  (i.e., where the data values are grouped into
  intervals, 0-9, 10-19, 20- 29, etc., and the
  frequencies of the intervals are shown).

19

• Graphic Representations of Data
• Another excellent way to clearly describe your
  data (especially for visually oriented learners)
  is to construct graphical representations of the
  data (i.e., pictorial representations of the data
  in two-dimensional space).  
• A bar graph uses vertical bars to represent the
  data. The height of the bars usually represent
  the frequencies for the categories shown on the X
  axis(i.e., the horizontal axis). (By the way, the
  Y axis is the vertical axis.)

20

• A line graph uses one or more lines to depict
  information about one or more variables.  
• A simple line graph might be used to show a trend
  over time (e.g., with the years on the X axis and
  the population sizes on the Y axis).
• Line graphs are used for many different purposes
  in research. For example, (GPA is on the X axis
  and frequency is on the Y axis)

21

• A scatterplot is used to depict the relationship
  between two quantitative variables.
• Typically, the independent or predictor variable
  is represented by the X axis (i.e., on the
  horizontal axis) and the dependent variable is
  represented by the Y axis (i.e., on the vertical
  axis).

22

• The relationship is not always positive
• Correlation coefficient range between -1 and 1
• Interpretation of Pearson r
• 1 highly positvely correlated
• -1 highly negatively correlated
• Close to zero, no correlation

23

• Correlation does not necessarily indicate
  causation
• .82 tells us that a person with an average score
  on the test will probably obtained an average
  score on other test

24

• How to Interpret the Values of Correlations.
• The correlation coefficient (r) represents the
  linear relationship between two variables. If the
  correlation coefficient is squared, then the
  resulting value (r2, the coefficient of
  determination) will represent the proportion of
  common variation in the two variables (i.e., the
  "strength" or "magnitude" of the relationship).
• In order to evaluate the correlation between
  variables, it is important to know this
  "magnitude" or "strength" as well as the
  significance of the correlation.

25

• Outliers.
• Outliers are atypical (by definition), infrequent
  observations.
• Outliers have a profound influence on the slope
  of the regression line and consequently on the
  value of the correlation coefficient.
• A single outlier is capable of considerably
  changing the slope of the regression line and,
  consequently, the value of the correlation, as
  demonstrated in the following example.

26
(No Transcript)
27

• Analyses for Comparison
• Nominal Data Chi-Square
• Interval Data t-Test
• Interval Data One-Way ANOVA
• Interval Data Factorial ANOVA
• Analyses for Association
• Interval Data Pearson Product-Moment Correlation
  (r)
• Nominal Data Phi Coefficient
• Ordinal Data Spearman Rank-Order Correlation

28
parametric Methods Non parametric Methods
t-test for independent samples Mann-Whitney U test
ANOVA/MANOVA (multiple groups) Kruskal-Wallis analysis of ranks and the Median test.
t-test for dependent samples (two variables measured in the same samplE) Sign test and Wilcoxon's matched pairs test
29

• t-test for independent samples
• Purpose, Assumptions.
• The t-test is the most commonly used method to
  evaluate the differences in means between two
  groups.
• For example, the t-test can be used to test for a
  difference in test scores between a group of
  patients who were given a drug and a control
  group who received a placebo.
• Theoretically, the t-test can be used even if the
  sample sizes are very small (e.g., as small as
  10 some researchers claim that even smaller n's
  are possible), as long as the variables are
  normally distributed within each group and the
  variation of scores in the two groups is not
  reliably different

30

• The normality assumption can be evaluated by
  looking at the distribution of the data (via
  histograms) or by performing a normality test.
• The equality of variances assumption can be
  verified with the F test, or you can use the more
  robust Levene's test.
• If these conditions are not met, then you can
  evaluate the differences in means between two
  groups using one of the nonparametric
  alternatives to the t- test (Nonparametrics).

31

• Independent sample t test

Mean N Std.Deviation Std. Error Mean
Talk Low stress High stress 42.20 22.07 15 15 24.97 27.14 6.45 7.01
Sx SD/v15
Standard deviation of the sample means
IV
DV
  F Sig. T Df Sig. (2-tailed Mean diff Std. error diff
Talk Equal variance assumed Equal variance not assumed .023 .881 2.43 2.430 28 27.808 .022 .022  .
In this case, variances are similar
Tested at a .05
Levenes test for equality of variance
You want a small F
Here you want variance to equal
The larger the F value the more dissimilar the
varainces are
32

• An independent t st was conducted to evaluate the
  hypothesis that students talk differently (amount
  of talkin) under different stress condition. The
  test was significant, t (28) 2.43, p .022.
  Students in high stress-condition talked less
  (M22.07 SD 27.14) than students in
  low-stressed condition (M45.20 SD 24.97)

33

• t-test for dependent samples (paired sampel
  t-test
• Test two groups of observations (that are to be
  compared) are based on the same sample of
  subjects who were tested twice (e.g., before and
  after a treatment )

Mean N Std.Deviation Std. Error Mean
PAY SECURITY 5.67 4.50 30 30 1.49 1.83 .27 .33
Sx SD/v30
Standard deviation of the sample means
34
  Mean Std. Dev. Std. Err. Lower Upper t df Sig. (2-tailed)
Pay- security 1.17 2.26 .41 .32 2.01 2.827 29  .008

• A paired-sample t test was conducted to evaluate
  whether employees were more concerned with pay or
  job security. The results indicated that the mean
  concern for pay (M 5.67, SD 1.49) was
  significantly greater than the mean concern for
  security (M 4.50, SD 1.83), t (29) 2.83, p
  .008.

35

• It was suggested (Marija J. Norusis) that
• When reporting your results, give the exact
  observed significance level. It will help the
  rader evaluate your findings
• Eg p .008, 8 chances in 1000 you would
  observe the difference between the two sample.
• Eg p .08 8 chances in 100 but you have set
  that you will only acet if it is 5 chances in
  100

36

• Pearson Chi-square.
• The Pearson Chi-square is the most common test
  for significance of the relationship between
  categorical variables.
• This measure is based on the fact that we can
  compute the expected frequencies in a two-way
  table (i.e., frequencies that we would expect if
  there was no relationship between the variables).
  
• For example, suppose we ask 20 males and 20
  females to choose between two brands of jeans
  (brands A and B).
• If there is no relationship between preference
  and gender, then we would expect about an equal
  number of choices of brand A and brand B for each
  sex.
• The Chi-square test becomes increasingly
  significant as the numbers deviate further from
  this expected pattern that is, the more this
  pattern of choices for males and females differs.

37

• The Goodness of Fit test used to find out if the
  population under study follow the distribution
  values
• Ho the population distribution is uniform, that
  is, each brand of cola drinks is prefered by an
  equal percentage of the population
• Ha the population distribution is not uniform,
  that is, each brand of cola drinks is not
  prefered by an equal percentage of the population

38
brand O E O-E (O-E)2 (O-E)2/E
A 50 60
B 65 60
C 45 60
D 70 60
E 70 60
Total 300 60
X 2 (df5) 9.18, let say the significant value
is 9.49, then Ho has to rejected and we cannot
say that cola brands are preferred by an equal
percentage of the population Df (r-1). (c-1)
39

• Test of independence we can test the
  realtionship between nominal variables)
• The data are obtained from a random sample
• We use count data (frequencies)
• We want to test whether perception of life is
  independent of gender or men and women find life
  equaly exciting

40
Life excitement male female
excited 300 384 684
Not excited 296 481 777
596 865 1461
Chi square 4.76, DF 1 p .0290
What can you conclude?