Data analysis

The mean, median and mode
Presenter: Huu Loc

The mean, median and mode are all valid
measures of central tendency but, under
different conditions, some measures of
central tendency become more
appropriate to use than others.

Mean
 The mean (or average) is the most
popular and well known measure of
central tendency. It can be used with
both discrete and continuous
data, although its use is most often
with continuous data.
 The mean is equal to the sum of all
the values in the data set divided by
the number of values in the data set.

So, if we have n values in a data set and
they have values x1, x2, ..., xn, then the
sample mean, usually denoted by
(pronounced x bar), is:

 The mean is essentially a model of your
data set. It is the value that is most
common.
 An important property of the mean is
that it includes every value in your data
set as part of the calculation. In
addition, the mean is the only measure
of central tendency where the sum of
the deviations of each value from the
mean is always zero.

Median

The median is the middle score for a set of
data that has been arranged in order of
magnitude. The median is less affected by
outliers and skewed data. In order to
calculate the median, suppose we have
the data below:

We first need to rearrange that data into
order of magnitude (smallest first):

Our median mark is the middle mark -
in this case 56 (highlighted in bold). It
is the middle mark because there are 5
scores before it and 5 scores after it.

Mode
The mode is the most frequent score in
our data set. On a histogram it
represents the highest bar in a bar
chart or histogram. You can, therefore,
sometimes consider the mode as
being the most popular option. An
example of a mode is presented
below:

Normally, the mode is used for
categorical data where we wish to know
which is the most common category as
illustrated below:

One of the problems with the mode is that it is
not unique, so it leaves us with problems when
we have two or more values that share the
highest frequency, such as below:

Summary of when to use the
mean, median and mode
Using the following summary table to know
what the best measure of central tendency is
with respect to the different types of variable.

MEASURES OF
DISPERSION
Presenter: Nguyen Ngoc Cam

Measures of Dispersion

Measure of central tendency give us good information about the scores in
our distribution.

However, we can have very different shapes to our distribution, yet have
the same central tendency.

Measures of dispersion or variability will give us information about the
spread of the scores in our distribution.

Are the scores clustered close together over a small portion of the scale, or
are the scores spread out over a large segment of the scale?

Main points:

1. Range

2. Standard Deviation

3. Variance

1. Range

The difference between the biggest and the
smallest number in the data of the group.

The range tells you how spread out the data
is.

1. Range

Problem:

1. It changes drastically with the magnitude of the extreme
scores

2. It’s an unstable measure  rarely used for statistical
analyses


Standard Deviation is the most frequently used
measure of variability.

It looks at the average variability of all the score
around the mean, all the scores are taken into
account.


The larger the Standard Deviation, the more
variability from the central point in the
distribution.

The smaller the Standard Deviation, the closer
the distribution is to the central point.


The SD tells us the standard of how far out from
the point of central tendency the individual
scores are distributed.

It tells us information that the mean doesn’t 
as important or even more important than the
mean

PAIRED T-TEST
Presenter: Tran Thi Ngan Giang

Introduction
• A paired t-test is used to compare two population
means where you have two samples in which
observations in one sample can be paired with
observations in the other sample.
• For example:
• A diagnostic test was made before studying a
particular module and then again after
completing the module. We want to find out if, in
general, our teaching leads to improvements in
students’ knowledge/skills.

First, we see the descriptive statistics
for both variables.

The post-test mean scores are higher.

Next, we see the correlation between
the two variables.

There is a strong positive correlation. People who
did well on the pre-test also did well on the post-
test.

Finally, we see the T, degrees of
freedom, and significance.
• Our significance is .053
• If the significance value is less
than .05, there is a significant
difference.
If the significance value is greater
than. 05, there is no significant
difference.
• Here, we see that the significance
value is approaching
significance, but it is not a
significant difference. There is no
difference between pre- and
post-test scores. Our test
preparation course did not help!

INDEPENDENT SAMPLES T-
TESTS

Presenter: Dinh Quoc Minh Dang

Outline

1. Introduction

2. Hypothesis for the independent t-test

3. What do you need to run an independent t-test?

4. Formula

5. Example (Calculating + Reporting)

Introduction

The independent t-test, also called the two sample t-test or student's t-test is
an inferential statistical test that determines whether there is a statistically
significant difference between the means in two unrelated groups.

Hypothesis for the independent t-test

The null hypothesis for the independent t-test is that the population means from the two
unrelated groups are equal:

H0: u1 = u2

In most cases, we are looking to see if we can show that we can reject the null hypothesis
and accept the alternative hypothesis, which is that the population means are not equal:

HA: u1 ≠ u2

To do this we need to set a significance level (alpha) that allows us to either reject or accept
the alternative hypothesis. Most commonly, this value is set at 0.05.

What do you need to run an independent t-test?

In order to run an independent t-test you need the following:

1. One independent, categorical variable that has two levels.

2. One dependent variable

Formula

M: mean (the average score of the group)

SD: Standard Deviation

N: number of scores in each group

Exp: Experimental Group

Con: Control Group

Reporting the Result of an Independent T-Test

When reporting the result of an independent t-test, you need to include the t-
statistic value, the degrees of freedom (df) and the significance value of the
test (P-value). The format of the test result is: t(df) = t-statistic, P =
significance value.

Example result (APA Style)

An independent samples T-test is presented the same as the one-sample t-test:

t(75) = 2.11, p = .02 (one –tailed), d = .48

Degrees
of
freedom
Value of Effect
statistic size if
Significance Include if test available
of statistic is one-tailed

Example: Survey respondents who were employed by the federal, state, or local
government had significantly higher socioeconomic indices (M = 55.42, SD =
19.25) than survey respondents who were employed by a private employer (M =
47.54, SD = 18.94) , t(255) = 2.363, p = .01 (one-tailed).

Analysis of Variance (ANOVA)

Presenter : Minh Sang

Introduction
We already learned about the chi square test
for independence, which is useful for data
that is measured at the nominal or ordinal
level of analysis.
If we have data measured at the interval
level, we can compare two or more
population groups in terms of their
population means using a technique called
analysis of variance, or ANOVA.

Completely randomized design
Population 1 Population 2….. Population k
Mean = 1 Mean = 2 …. Mean = k
Variance= 12 Variance= 22 … Variance = k2

We want to know something about how the
populations compare. Do they have the same
mean? We can collect random samples from each
population, which gives us the following data.

Mean = M1 Mean = M2 ..… Mean = Mk
Variance=s12 Variance=s22 …. Variance = sk2
N1 cases N2 cases …. Nk cases

Suppose we want to compare 3 college majors in a
business school by the average annual income
people make 2 years after graduation. We collect
the following data (in $1000s) based on random
surveys.

Accounting Marketing Finance
27 23 48
22 36 35
33 27 46
25 44 36
38 39 28
29 32 29

Can the dean conclude that there are
differences among the major’s incomes?
H o: 1 = 2 = 3
HA: 1 2 3

In this problem we must take into account:
1) The variance between samples, or the actual
differences by major. This is called the sum of
squares for treatment (SST).

2) The variance within samples, or the
variance of incomes within a single major.
This is called the sum of squares for error
(SSE).
Recall that when we sample, there will always
be a chance of getting something different
than the population. We account for this
through #2, or the SSE.

F-Statistic
For this test, we will calculate a F
statistic, which is used to compare
variances.
F = SST/(k-1)
SSE/(n-k)
SST=sum of squares for treatment
SSE=sum of squares for error
k = the number of populations
N = total sample size

F-statistic
Intuitively, the F statistic is:
F = explained variance
unexplained variance
Explained variance is the difference between
majors
Unexplained variance is the difference based
on random sampling for each group (see
Figure 10-1, page 327)

Calculating SST
SST = ni(Mi - )2
= grand mean or = Mi/k or the sum of
all values for all groups divided by total
sample size
Mi = mean for each sample
k= the number of populations

Calculating SST
By major
Accounting M1=29, n1=6
Marketing M2=33.5, n2=6
Finance M3=37, n3=6
= (29+33.5+37)/3 = 33.17
SST = (6)(29-33.17)2 + (6)(33.5-33.17)2 +
(6)(37-33.17)2 = 193

Calculating SST
Note that when M1 = M2 = M3, then SST=0
which would support the null hypothesis.
In this example, the samples are of equal size,
but we can also run this analysis with
samples of varying size also.

Calculating SSE
SSE = (Xit – Mi)2
In other words, it is just the variance for each sample
added together.
SSE = (X1t – M1)2 + (X2t – M2)2 +
(X3t – M3)2
SSE = [(27-29)2 + (22-29)2 +…+ (29-29)2]
+ [(23-33.5)2 + (36-33.5)2 +…]
+ [(48-37)2 + (35-37)2 +…+ (29-37)2]
SSE = 819.5

Statistical Output
When you estimate this information in a computer
program, it will typically be presented in a table as
follows:
Source of df Sum of Mean F-ratio
Variation squares squares
Treatment k-1 SST MST=SST/(k-1) F=MST
Error n-k SSE MSE=SSE/(n-k) MSE
Total n-1 SS=SST+SSE

Calculating F for our example
F = 193/2
819.5/15
F = 1.77
Our calculated F is compared to the critical
value using the F-distribution with
F , k-1, n-k degrees of freedom
k-1 (numerator df)
n-k (denominator df)

The Results
For 95% confidence ( =.05), our critical F is
3.68 (averaging across the values at 14 and
16
In this case, 1.77 < 3.68 so we must accept the
null hypothesis.
The dean is puzzled by these results because
just by eyeballing the data, it looks like
finance majors make more money.

The Results
Many other factors may determine the salary
level, such as GPA. The dean decides to
collect new data selecting one student
randomly from each major with the
following average grades.

New data
Average Accounting Marketing Finance M(b)
A+ 41 45 51 M(b1)=45.67
A 36 38 45 M(b2)=39.67
B+ 27 33 31 M(b3)=30.83
B 32 29 35 M(b4)=32
C+ 26 31 32 M(b5)=29.67
C 23 25 27 M(b6)=25
M(t)1=30.83 M(t)2=33.5 M(t)3=36.83

= 33.72

Randomized Block Design
Now the data in the 3 samples are not
independent, they are matched by GPA
levels. Just like before, matched samples
are superior to unmatched samples because
they provide more information. In this
case, we have added a factor that may
account for some of the SSE.

Two way ANOVA
Now SS(total) = SST + SSB + SSE
Where SSB = the variability among blocks,
where a block is a matched group of
observations from each of the populations
We can calculate a two-way ANOVA to test
our null hypothesis. We will talk about this
next week.

Data analysis

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