Statistics101: Numerical Measures

Intro to Statistics for non-statisticians (delivered as part of Six Sigma Green Belt training)

1. Zahid Mian
Part of the Brown-bag Series

2. Population
Sample
Variable
Statistic
Sample
Skew

3. Mean
Median
Mode
Range
Percentile
Variance
Standard Deviation
Covariance
Correlation Coefficient
Skewness

4. Why do we care about Central Tendency?
What is most valuable to you:
Average price of home in a neighborhood
Median price of home …
Range of prices …
Mode …
What does it say about neighborhood if:
Average price is $500K
Median price is $350K
Range is $750K

5. ??????for Population ??????=
??????
??????
 ??????for Sample ??????=
??????
�
N = 1,2,3,4,4,5,5,5,5,6
Mean = 4

6. The “middle” value from sorted list
??????��??????????????????=
�+1
2
�ℎ
term
Data: 1,2,3,4,4,5,5,5,5,6 Median: 4.5
Data: 1,2,3,4,4,5,5,5,5,6,7 Median: 5

7. The number that occurs the most
Data: 1,2,3,4,4,5,5,5,5,6,7
Mode: 5 (appears 4 times)
> table(c(1,2,3,4,4,5,5,5,5,6,7))
1 2 3 4 56 7
1 1 1 2 41 1

8. Cuts off data by npercent
Quiz Scores:
67,72,88,82,80,90,95,60,77,89,99,85,77
What is the score that cuts off 30% of all
scores? 50%?
quantile(c(67,72,88,82,80,90,95,60,77,89,99,85,77 ), c(.3, .5))
30% 50%
77 82
30% of all scores were 77 or below; 50% of all scores were 82 or
below

9. Graph showing data within quartiles
Max Value
Min Value
Median
Q3 (75%)
Q1 (25%)

10. How to see the “dispersion” of data
Sets of quiz scores for different classes:
Set1: 80, 79, 80, 81, 80, 80, 79, 79
Set2: 75, 100, 60, 100, 100, 75, 75, 60
Just by looking at the number you should be
able to state that Set2 is more dispersed
standard deviation measures the dispersion
around the mean
Other measures include range and variance

11. difference between the highest and lowest
values in the data
Data: 1,2,3,4,4,5,5,5,5,6,7
Range: 6 (7-1)

12. Measures dispersion around the mean (how
far from the normal)
??????
2
=
(�−??????)
2
�
Steps to calculate population variance (??????
2
)
Calculate mean
For each number in set
▪subtract the mean
▪Square the result (why square it?)
Get the average of differences

13. Data:
67,72,88,82,80,90,95,6
0,77,89,99,85,77
Large variance
indicates data is
spread out; small
indicates data is close
to mean.
For Sample (note n-1)
�
2
=
(�− �)
2
�−1
Mean: 81.62
(67-81.62)
2
= 213.74
(72-81.62)
2
= 92.54
…
(72-81.62)
2
= 21.34
Average of 213.74,
92.54,…, 21.74
??????
2
= 123.09

14. Simply the square root of variance
??????=

??????
(�
??????
−??????)
2
??????
??????
2
= 123.09
??????= 11.09
??????is useful in determining what is “normal”
The mean of scores was 81.62, so most scores
are within 1 ??????(+/-11.09). All scores are within
2 ??????(+/-22.18).

15. 1
??????
Mean
2
??????

16. Measures how two variables (x,y) are linearly
related. Positive value indicates linear relation.
Test Scores (x):
67,72,88,82,80,90,95,60,77,89,99,85,77
Study Time (y):
30,45,80,85,75,85,120,30,45,75,85,110,40
Is there a relationship between Test Scores and
Time spent studying?
??????
��= 269.43
What if everyone studied for 30 minutes?
??????
��= 0 (so no linear relation)

17. A normalized measurement of how two
variables are linearly related.
Sample: �
��=
�
��
�
��
�
Population: ??????
��=
??????��
??????
�??????
�
From Previous Example: ??????
��= 0.83 (the closer
this value to 1, the stronger the relationship)

18. Intuitively we would say there is no relation
But be careful …
Let’s say I have data for sales of ice vs. temp
> cor(temps, sales)
[1] -0.001413245
Correlation Coefficient is nearly 0, so no
relation, right?

19. The scatter plot
clearly shows a
strong relationship
between sales and
temps. Maybe
when it’s too hot
people just don’t
want to leave the
house.
Always visualize data!

20. Identical simple statistical properties—so always Visualize!

21. Measure of “symmetry” of
data. Negative value
indicates mean is less than
median (left skewed).
Positive value indicates
mean is larger than
median (right skewed).
> skewness(scores)
[1] -0.302365

22. Data:
60,62,62,62,65,65,65,
75,82,96,99,100,100
> skewness(scores)
[1] 0.4652821
This means a few
students did really well
and lifted the overall
mean score.

24. Mean, Median, Mode exactly at center
99.999% of all data within 3 ??????of mean
Important for making inferences
Test scores are generally normal distribution
Height of humans follow normal distribution
Need to be careful not to apply normal
distribution rules against non-normal data
https://en.wikipedia.org/wiki/Normality_test

25. z-Score indicates how far above or below the mean
a given score in the distribution is
Scenario: Which exam did Scott do better?
Scott got a 65/100 on Exam1; ??????is 60; ??????is 10
Scott got a 42/200 on Exam2; ??????is 37; ??????is 5
First, need to standardize scores (Exam1 is out of
100; Exam2 out of 200)
This standardization is the z-score
??????=
�??????������−��??????�
��??????��??????�����????????????�??????��
or ??????=
??????−??????
??????

26. z of -1.5 means student
scored 1.5 standard
deviations below the
mean
In the case of test scores,
positive numbers are good
Less than 10% scored worse
Which score marks the
97
th
percentile?
What percentage of
population scored
between score1 and
score2 (say 75 and 90)?

27. Measurements of Central Tendency and
Variability are critical to study of statistics
Central Tendency tries to provide information
about the “central” value of your set
Variability tries to provide information about
the dispersion of data in your set
Covariance tries to provide information about
how two variables are related
z-Scores are useful with a normal distribution