STATISTICS 1 – DESCRIPTION AND INFERENCE
WEEK 1: Distributions, means and deviation
Variable: anything that can be measured and can differ across entities or across time
1. Independent variables: Cause (x)
2. Dependent variables: Outcome (y)
Levels of measurement (categorical)
Nature of information within values assigned to variables
1. Nominal
• Two or more exclusive categories
• No natural order
• No arithmetic operations possible (subtraction, “equal to”, “greater than”)
• Frequency (mode)
• Occupation, political party affiliation, favorite football club
2. Ordinal
• Clear ordering of the variables
• Low <-> high; little <-> much; small <-> large
• Spacing between values not the same across the levels of the variables
• Level of agreement, level of education, political interest, trust in government
3. Interval
• Difference between two values meaningful
• Arbitrary/meaningless zero
• Temperature, pH (more examples in natural sciences)
4. Ratio
• Like interval but meaningful zero
• Height, weight, salary, Kelvin, number of international conflicts
Continuous variables – interval and ratio
Continuous variables (interval-ratio) are continuous but can be discrete.
o Continuous interval-ratio variables
• Can be measured to any level of precision
• Height or weight: 75.329242… cm
o Discrete interval-ratio variables
• Can take only certain, countable values (usually whole numbers)
• Points in an exam or number of car accidents at an intersection
• Discrete variables can be measured in discrete terms (height in cm)
,What can you do?
Possible to… Nominal Ordinal Interval Ratio
Frequency distribution? YES YES YES YES
Median and percentiles? NO YES YES YES
Add or subtract? NO NO YES YES
Mean, standard NO NO YES YES
deviation, standard error
of the mean?
Ratio, or coefficient of NO NO NO YES
variation?
I. DISTRIBUTION
To show how data values are distributed in relation to other values.
> Frequency distribution: the distribution of a statistical data set to show all the possible
values (or intervals) of the data and how often they occur.
Measure of central tendency
A value that attempts to describe a set of data by identifying the central position within that
set of data
Level of measurement Measures of central tendency
Nominal Mode
Ordinal Median + Mode
Interval and ratio Mean + Median + Mode
MODE: the most frequent score in a data set (there can be several modes)
MEDIAN: the middle score for a set of data that has been arranged in order of magnitude
o Odd number of scores: Arrange by magnitude to find median.
o Even number of scores: Arrange by magnitude, add 2 middle scores, divide by 2
MEAN: the average of the numbers
1. Calculate the sum of all values of x
2. Divide by the total number of observations (n)
&
Σ ' = 1)*
!" =
&
Mean is sensitive to extreme values/outliers
When there are extreme values, the median may be more useful
,Measure of dispersion
How stretched or squeezed is the distribution?
Level of measurement Measures of dispersion
Nominal No measure of dispersion possible
Ordinal Range, inter-quartile range
Interval and ratio Range, inter-quartile range,
variance/standard variation
RANGE: maximum - minimum
INTERQUARTILE RANGE: The range of the middle 50% of the data.
Calculate by subtracting the first quartile from the third quartile.
Even set of numbers?
1. Split the data in two halves
2. Find the median of the lower half
3. Find the median of the upper half
4. Calculate IQR = 57 – 35 = 22
VARIANCE AND STANDARD DEVIATION
Measure the spread using all data
DEVIANCE: How much does each value deviate from the mean?
+,-'.&/, = )* − )̅
Total deviance (sum of all deviances) is always zero.
&
2 = 1()* − )̅ )
'
, EXAMPLE: Number of points of students in a test
Adam Abby Dan Elsa Isan Liz Max Pat Tom Uma Zev
14 35 45 55 55 56 57 65 87 89 92
Step 1. Calculate the MEAN (column 2)
∑6*78 )*
!" =
&
89:;<:9<:<<:<<:<=:<>:=<:?>:?@:@A =<B
!" = 88
= 88
= 59.09
Step 2. Calculate all DEVIANCES (column 3)
Step 3. Square all the deviances (errors) (column 4) – SQUARED ERRORS
Step 4. Add the squared deviances – SUM OF SQUARED ERRORS (SS)
6
∑ = 1()* − )̅ )A = 5650.91
*
x )̅ ()* − )̅ ) ()* − )̅ )A
14 59.09 14 - 59.09 = -45.09 (−45.09)A = 2033.19
35 59.09 35 - 59.09 = -24.09 (−24.09)A = 580.37
45 59.09 45 - 59.09 = -14.09 (−14.09)A = 198.55
55 59.09 55 - 59.09 = -4.09 (−4.09)A = 16.74
55 59.09 55 - 59.09 = -4.09 (−4.09)A = 16.74
56 59.09 56 - 59.09 = -3.09 (−3.09)A = 9.55
57 59.09 57 - 59.09 = -2.09 (−2.09)A = 4.37
65 59.09 65 - 59.09 = 5.91 (5.91)A = 34.92
87 59.09 87 - 59.09 = 27.91 (27.91)A = 778.92
89 59.09 89 - 59.09 = 29.91 (29.91)A = 894.55
92 59.09 92 - 59.09 = 32.91 (32.91)A = 1038.01
5650.91
Step 5. Calculate the VARIANCE (K A )
LL ∑6*78()* − )̅ )A
KA = =
M−1 M−1
LL 5650.91 5650.91
KA = = = = OPO. QR
M−1 11 − 1 10
Step 6. Calculate STANDARD DEVIATION (s)
K = SK A
K = SK A = √565.09 = UV. WW
WEEK 1: Distributions, means and deviation
Variable: anything that can be measured and can differ across entities or across time
1. Independent variables: Cause (x)
2. Dependent variables: Outcome (y)
Levels of measurement (categorical)
Nature of information within values assigned to variables
1. Nominal
• Two or more exclusive categories
• No natural order
• No arithmetic operations possible (subtraction, “equal to”, “greater than”)
• Frequency (mode)
• Occupation, political party affiliation, favorite football club
2. Ordinal
• Clear ordering of the variables
• Low <-> high; little <-> much; small <-> large
• Spacing between values not the same across the levels of the variables
• Level of agreement, level of education, political interest, trust in government
3. Interval
• Difference between two values meaningful
• Arbitrary/meaningless zero
• Temperature, pH (more examples in natural sciences)
4. Ratio
• Like interval but meaningful zero
• Height, weight, salary, Kelvin, number of international conflicts
Continuous variables – interval and ratio
Continuous variables (interval-ratio) are continuous but can be discrete.
o Continuous interval-ratio variables
• Can be measured to any level of precision
• Height or weight: 75.329242… cm
o Discrete interval-ratio variables
• Can take only certain, countable values (usually whole numbers)
• Points in an exam or number of car accidents at an intersection
• Discrete variables can be measured in discrete terms (height in cm)
,What can you do?
Possible to… Nominal Ordinal Interval Ratio
Frequency distribution? YES YES YES YES
Median and percentiles? NO YES YES YES
Add or subtract? NO NO YES YES
Mean, standard NO NO YES YES
deviation, standard error
of the mean?
Ratio, or coefficient of NO NO NO YES
variation?
I. DISTRIBUTION
To show how data values are distributed in relation to other values.
> Frequency distribution: the distribution of a statistical data set to show all the possible
values (or intervals) of the data and how often they occur.
Measure of central tendency
A value that attempts to describe a set of data by identifying the central position within that
set of data
Level of measurement Measures of central tendency
Nominal Mode
Ordinal Median + Mode
Interval and ratio Mean + Median + Mode
MODE: the most frequent score in a data set (there can be several modes)
MEDIAN: the middle score for a set of data that has been arranged in order of magnitude
o Odd number of scores: Arrange by magnitude to find median.
o Even number of scores: Arrange by magnitude, add 2 middle scores, divide by 2
MEAN: the average of the numbers
1. Calculate the sum of all values of x
2. Divide by the total number of observations (n)
&
Σ ' = 1)*
!" =
&
Mean is sensitive to extreme values/outliers
When there are extreme values, the median may be more useful
,Measure of dispersion
How stretched or squeezed is the distribution?
Level of measurement Measures of dispersion
Nominal No measure of dispersion possible
Ordinal Range, inter-quartile range
Interval and ratio Range, inter-quartile range,
variance/standard variation
RANGE: maximum - minimum
INTERQUARTILE RANGE: The range of the middle 50% of the data.
Calculate by subtracting the first quartile from the third quartile.
Even set of numbers?
1. Split the data in two halves
2. Find the median of the lower half
3. Find the median of the upper half
4. Calculate IQR = 57 – 35 = 22
VARIANCE AND STANDARD DEVIATION
Measure the spread using all data
DEVIANCE: How much does each value deviate from the mean?
+,-'.&/, = )* − )̅
Total deviance (sum of all deviances) is always zero.
&
2 = 1()* − )̅ )
'
, EXAMPLE: Number of points of students in a test
Adam Abby Dan Elsa Isan Liz Max Pat Tom Uma Zev
14 35 45 55 55 56 57 65 87 89 92
Step 1. Calculate the MEAN (column 2)
∑6*78 )*
!" =
&
89:;<:9<:<<:<<:<=:<>:=<:?>:?@:@A =<B
!" = 88
= 88
= 59.09
Step 2. Calculate all DEVIANCES (column 3)
Step 3. Square all the deviances (errors) (column 4) – SQUARED ERRORS
Step 4. Add the squared deviances – SUM OF SQUARED ERRORS (SS)
6
∑ = 1()* − )̅ )A = 5650.91
*
x )̅ ()* − )̅ ) ()* − )̅ )A
14 59.09 14 - 59.09 = -45.09 (−45.09)A = 2033.19
35 59.09 35 - 59.09 = -24.09 (−24.09)A = 580.37
45 59.09 45 - 59.09 = -14.09 (−14.09)A = 198.55
55 59.09 55 - 59.09 = -4.09 (−4.09)A = 16.74
55 59.09 55 - 59.09 = -4.09 (−4.09)A = 16.74
56 59.09 56 - 59.09 = -3.09 (−3.09)A = 9.55
57 59.09 57 - 59.09 = -2.09 (−2.09)A = 4.37
65 59.09 65 - 59.09 = 5.91 (5.91)A = 34.92
87 59.09 87 - 59.09 = 27.91 (27.91)A = 778.92
89 59.09 89 - 59.09 = 29.91 (29.91)A = 894.55
92 59.09 92 - 59.09 = 32.91 (32.91)A = 1038.01
5650.91
Step 5. Calculate the VARIANCE (K A )
LL ∑6*78()* − )̅ )A
KA = =
M−1 M−1
LL 5650.91 5650.91
KA = = = = OPO. QR
M−1 11 − 1 10
Step 6. Calculate STANDARD DEVIATION (s)
K = SK A
K = SK A = √565.09 = UV. WW
