Basic stat review

Level of Measurement:
Nominal, Ordinal, Ratio, Interval
Frequency tables to distributions
Central Tendency: Mode, Median, Mean
Dispersion: Variance, Standard Deviation

 Nominal – categorical data ; numbers represent
labels or categories of data but have no
quantitative value e.g. demographic data
 Ordinal – ranked/ordered data ; numbers rank or
order people or objects based on the attribute
being measured
 Interval – points on a scale are an equal distance
apart ; scale does not contain an absolute zero
point
 Ratio – points on a scale are an equal distance
apart and there is an absolute zero point

 Simple depiction of all the data
 Graphic — easy to understand
 Problems
 Not always precisely measured
 Not summarized in one number or datum

Observation Frequency
65 1
70 2
75 3
80 4
85 3
90 2
95 1

4

3
Frequency
2

1

65 70 75 80 85 90 95

Test Score

Two key characteristics of a frequency distribution
are especially important when summarizing
data or when making a prediction from one set
of results to another:
 Central Tendency
 What is in the “Middle”?
 What is most common?
 What would we use to predict?
 Dispersion
 How Spread out is the distribution?
 What Shape is it?

Three measures of central tendency are commonly
used in statistical analysis - the mode, the median,
and the mean
Each measure is designed to represent a typical
score
The choice of which measure to use depends on:
 the shape of the distribution (whether normal or

skewed), and
 the variable’s “level of measurement” (data are

nominal, ordinal or interval).

 Nominal variables Mode

 Ordinal variables Median

 Ratio/Interval level variables Mean

 Most Common Outcome

Male Female

 Middle-most Value
 50% of observations are above the Median, 50%
are below it
 The difference in magnitude between the
observations does not matter
 Therefore, it is not sensitive to outliers

• first you rank order the values of X from low to
high:  85, 94, 94, 96, 96, 96, 96, 97, 97, 98
• then count number of observations = 10.
• divide by 2 to get the middle score  the 5
score
here 96 is the middle score

 Most common measure of central tendency
 Best for making predictions
 Applicable under two conditions:
1. scores are measured at the interval level, and
2. distribution is more or less normal [symmetrical].
 Symbolized as:

X
for the mean of a sample
 μ for the mean of a population

 X = (Σ X) / N
 If X = {3, 5, 10, 4, 3}
X = (3 + 5 + 10 + 4 + 3) / 5
= 25 / 5
= 5

IF THE DISTRIBUTION IS
NORMAL

Mean is the best measure of central
tendency
 Most scores “bunched up” in
middle
 Extreme scores less frequent 
don’t move mean around.
But, central tendency doesn’t tell us
everything !

Dispersion/Deviation/Spread tells us a lot about
how a variable is distributed.
We are most interested in Standard Deviations (σ)
and Variance (σ2)

 Consider these means for weekly candy bar
consumption.

X = {7, 8, 6, 7, 7, 6, 8, 7} X = {12, 2, 0, 14, 10, 9, 5, 4}
X = (7+8+6+7+7+6+8+7)/8 X = (12+2+0+14+10+9+5+4)/8
X=7 X=7

What is the difference?

Once you determine that the variable of interest is
normally distributed, ideally by producing a
histogram of the scores, the next question to be
asked is its dispersion: how spread out are the
scores around the mean.
Dispersion is a key concept in statistical thinking.
The basic question being asked is how much do
the scores deviate around the Mean? The more
“bunched up” around the mean the better the
scores are.

. If every X were very close to the Mean, the
mean would be a very good predictor.
If the distribution is very sharply peaked then
the mean is a good measure of central
tendency and if you were to use the mean to
make predictions you would be right or
close much of the time.

The key concept for describing normal distributions
and making predictions from them is called
deviation from the mean.
We could just calculate the average distance between
each observation and the mean.
 We must take the absolute value of the distance,
otherwise they would just cancel out to zero!
Formula:
| X − Xi |
∑ n

Data: X = {6, 10, 5, 4, 9, 8} X = 42 / 6 = 7

X – Xi Abs. Dev.
1. Compute X (Average)
7–6 1 2. Compute X – X and take
7 – 10 3 the Absolute Value to get
Absolute Deviations
7–5 2
3. Sum the Absolute
7–4 3 Deviations
7–9 2 4. Divide the sum of the
absolute deviations by N
7–8 1
Total: 12 12 / 6 = 2

 Instead of taking the absolute value, we square
the deviations from the mean. This yields a
positive value.
 This will result in measures we call the

Variance and the Standard Deviation
Sample- Population-
s: Standard Deviation σ: Standard Deviation
s2: Variance σ2: Variance

Formulae:

Variance: Standard Deviation:

s =
2 ∑(X − Xi ) 2
s=
∑(X − X ) i
2

N N

Data: X = {6, 10, 5, 4, 9, 8}; N=6
Mean:
X X−X (X − X ) 2

X=
∑X =
42
=7
6 -1 1 N 6
10 3 9 Variance:
5 -2 4
s =
2 ∑ ( X − X )2
=
28
= 4.67
4 -3 9 N 6
9 2 4 Standard Deviation:
8 1 1 s = s 2 = 4.67 = 2.16
Total: 42 Total: 28

Basic stat review

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