Psyc 235:Introduction to

Statistics

Lecture Format• New Content/Conceptual Info• Questions & Work through problems

What you should have accomplished so far…

• ALEKS account set up• completed first assessment• Worked through first section of

material• Spent 5+ hours on ALEKS• Watched the video “What is

statistics?”

Any questions/problems so far?

From Last week:

• Definition of Statistics…

C Collecting …

O Organizing …

D Displaying …

I Interpreting …

A Analyzing …

Data

What is Data?

• Data is the generic term for numerical information that has been obtained on a set of objects/individuals etc.

• Variable: Some characteristic of the objects/individuals (e.g.,

height)• Data:

the values of a variable for a certain set of objects/individuals

Two branches of statistics:

Descriptive StatisticsDescribes a given set of data you have.

Inferential StatisticsGiven the data you have about these people,does this say anything about other people?

Today: Descriptive Statistics

• Graphical Presentations of Distributions Histograms Frequency Polygons Cumulative Distributions Box-and-whisker plots

• Descriptive Measures of Data Measures of Central Tendency Measures of Dispersion

Organizing Data

• Data from last week• Frequency Table

Time Awake Number of Students6:30-7:00 17:00-7:30 17:30-8:00 38:00-8:30 28:30-9:00 49:00-9:30 59:30-10:00 710:00-10:30 410:30-11:00 3

6:557

7:307:307:45

88:258:308:458:458:50

999

9:159:259:309:309:309:309:309:459:45

1010

10:1510:2510:3010:4510:50

Histograms

Note: Use Histogram to note patterns in data. (Skew, etc.)

0

1

2

3

4

5

6

7

8

6:30-7:00

7:00-7:30

7:30-8:00

8:00-8:30

8:30-9:00

9:00-9:30

9:30-10:00

10:00-10:30

10:30-11:00

Wake-Up Time

Number of Students

Frequency Polygon

Time Awake Number of Students

Frequency

6:30-7:00 1 0.0333

7:00-7:30 1 0.0333

7:30-8:00 3 0.1

8:00-8:30 2 0.0667

8:30-9:00 4 0.1333

9:00-9:30 5 0.1667

9:30-10:00 7 0.2333

10:00-10:30 4 0.1333

10:30-11:00 3 0.1

Total 30 1

0

0.05

0.1

0.15

0.2

0.25

6:30-7:00

7:00-7:30

7:30-8:00

8:00-8:30

8:30-9:00

9:00-9:30

9:30-10:00

10:00-10:30

10:30-11:00

Time Awake

Proportion of Students

Cumulative Frequency

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

6:30-7:00

7:00-7:30

7:30-8:00

8:00-8:30

8:30-9:00

9:00-9:30

9:30-10:00

10:00-10:30

10:30-11:00

Time Awake

Time Awake Number of Students Frequency Cumulative6:30-7:00 1 0.03333333 0.03337:00-7:30 1 0.03333333 0.06677:30-8:00 3 0.1 0.16678:00-8:30 2 0.06666667 0.23338:30-9:00 4 0.13333333 0.36679:00-9:30 5 0.16666667 0.53339:30-10:00 7 0.23333333 0.766710:00-10:30 4 0.13333333 0.900010:30-11:00 3 0.1 1.0000Total 30 1

Box and Whisker Plots

• Graphical representation of the 4 quartiles, (e.g. data is split into 4 equally sized groups)

• If there are an even number of observations, let the “top” be the top half, and let the “bottom” be the bottom half.

• If there are an odd number of observations, let the “top” be everything above the median and the “bottom” be everything below the median.

• The first quartile is the “median of the bottom”. The third quartile is the “median of the top”.

Box-and-Whisker Example

6:557

7:307:307:45

88:258:308:458:458:50

999

9:159:259:309:309:309:309:309:459:45

1010

10:1510:2510:3010:4510:50

Median: 9:201st Quartile: 8:303rd Quartile: 9:45

Again, Note the information you can obtain by looking at this graphical representation of the data

Graphical Presentations of Data

• Listed Data: All data available

• Frequency Table:Data frequency for each cell is available

• Histograms: Data frequency for each bin is available

• Polygons: Data frequency for each bin is available

• Box-and-whisker plots:Summary info and data range available

• Often: Just summarize key features of the distribution.

Less And Less Information

Describing Distributions

Summary Measures

• Measures of Central Tendency “Average”, “Location”, “Center” of the distribution.

• Measures of Dispersion “Spread”, “Variability” of the distribution.

Summary Measures

• Measures of Central Tendency “Average”, “Location”, “Center” of the distribution.

• Measures of Dispersion “Spread”, “Variability” of the distribution.

Measures of Central Tendency

• Mean • Median• Mode

• May already be familiar with these concepts, but I want you to think of them in relation to describing data.

Mode

• Most frequent observation or observation class

• There can be several distinct modes• “Best guess” in single shot guessing

game 12

35

5

19ABCD

Mode (example data)

6:557

7:307:307:45

88:258:308:458:458:50

999

9:159:259:309:309:309:309:309:459:45

1010

10:1510:2510:3010:4510:50

Mode?

9:30

Median

• Any value M for which at least 50% of all observations are at or above M and at least 50% are at or below M.

• Resistant measure of central tendency (not heavily influenced by extreme values)

Calculating the Median

Order all observations from smallest to largest.

If the number of observations is odd, it is the “middle” object, namely the [(n+1)/2]th observation.For n = 61, it is the 31st

If the number of observations is even then, to get a unique value, take the average of the (n/2)th and the (n/2 +1)th observation. For = 60, it is the average of the 30th and the 31st observation.

Median (example data)

6:557

7:307:307:45

88:258:308:458:458:50

999

9:159:259:309:309:309:309:309:459:45

1010

10:1510:2510:3010:4510:50

Since there are an even number of data pointsTake the average of the middle two values.

,

Mean

• Sum up all observations (say, n many) and divide the total by n.

• Extreme values strongly influence the mean

• Mean as the center of the value in a distribution (center of gravity)

Calculating the mean

• Suppose that we collect n many observations

• Let denote the individual observations.

nXXXX ,...,,, 321

Mean • Sum up all observations (say, n many) and divide the total by n.

( )nn XXXnn

XXXX +++=

+++= ...

1...21

21Mean

Mathematical Notation

( )

∑∑ ==

+++=+++

=

ii

nn

Xnn

X

XXXnn

XXXX

1

...1...

2121Mean

n

n

ii XXXX +++=∑

=

...211

∑= iX

Mean (example data)

6:55 6.927 7

7:30 7.57:30 7.57:45 7.75

8 88:25 8.428:30 8.58:45 8.758:45 8.758:50 8.83

9 99 99 9

9:15 9.259:25 9.429:30 9.59:30 9.59:30 9.59:30 9.59:30 9.59:45 9.759:45 9.75

10 1010 10

10:15 10.2510:25 10.4210:30 10.510:45 10.7510:50 10.83

∑X = 273.34

X = 273.34 / 30 = 9.11

Transform back into time scale: ≈ 9:06

A few notes about summation, and implications for calculation of the mean

naaaa =+++ ...

n

naan

i

=∑=1

anaa n

n

in ==∑

=

1

1

1

0123456789

10

1 2 3 4 5

Mean

If all data has the same value, a, then the mean value is also a.

naan

i

=∑=1

because:

Multiplying all values by a constant

∑∑==

=n

ii

n

ii XaaX

11

( )nn XXXaaXaXaX +++=+++ ...... 2121

If we multiply each observationby 2, then we obtain a newdistribution with a different shape

A multiplying constant affects the mean

(and the “spread”)

XXXn

iin

n

iin 222

1

1

1

1 == ∑∑==

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

Adding a constant to all values

naXXX

aXaXaX

n

n

++++=++++++

)...()(...)()(

21

21

naXaXn

ii

n

ii +⎟

⎠

⎞⎜⎝

⎛=+ ∑∑

== 11

)(

1 2 3 4 5 6 7 8 9 10

If we add the constant 5to each observation,then we obtain a newdistribution that is shiftedto the right by 5 units

A shift affects the mean(but not the “spread”)

55

)5(

1

1

1

1

1

+=+⎟⎠

⎞⎜⎝

⎛=

+

∑

∑

=

=

XnX

X

n

n

iin

n

iin

1 2 3 4 5 6 7 8 9 10

Combining two variables

)...()...(

)(...)()(

2121

2211

nn

nn

YYYXXX

YXYXYX

+++++++=++++++

⎟⎠

⎞⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛=+ ∑∑∑

===

n

ii

n

ii

n

iii YXYX

111

)(

Adding two variables

⎟⎠

⎞⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛=+ ∑∑∑

===

n

ii

n

ii

n

iii YXYX

111

)(

YXYXYXn

iin

n

iin

n

iiin +=⎟

⎠

⎞⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛=+ ∑∑∑

=== 1

1

1

1

1

1 )(

The mean of the sum of two variables is the sum of their means

Measures of Dispersion

• Population Standard Deviation• Sample Standard Deviation

If we want to know how much the values vary around the

mean….

( ) ( ) ( )( )∑ −=

−++−+−

XX

XXXXXX

i

n...21

We could calculate how much each value varies from the mean…

Because of the way we calculate the mean, this formula gives zero no matter what data you have!

Population Standard Deviation

( ) ( ) ( )1

...22

2

2

12

−

−+−+−=

n

XXXXXXs n

( ) ( ) ( )1

...22

2

2

1

−

−+−+−=

n

XXXXXXs n

• Variance

• Standard Deviation

S

S

Sample Standard Deviation

• Variance

• Standard Deviation

( ) ( ) ( )1

...22

2

2

12

−

−+−+−=

n

XXXXXXs n

( ) ( ) ( )1

...22

2

2

1

−

−+−+−=

n

XXXXXXs n

There are n-1 “degrees of freedom”(If you know the mean and n-1 observationsthen you can figure out the n’th observation)

Computational Formulas

• Note that there are computational formulas for the standard deviation.

• Look for them in ALEKS and write them down.

• Remember you can bring notes to your assessments

For Next Week…

• Keep working on ALEKS• Finish the descriptive statistics section• Watch the second video• If you can, start probability section

before Jason’s lecture next week.

• Remember: Office Hours and Lab are always available for you.