statistics what is statistics? where are statistics used?

25
Statistics What is statistics? Where are statistics used?

Upload: eugene-bailey

Post on 18-Jan-2016

279 views

Category:

Documents


9 download

TRANSCRIPT

Page 1: Statistics What is statistics? Where are statistics used?

Statistics

What is statistics?

Where are statistics used?

Page 2: Statistics What is statistics? Where are statistics used?

What is Statistics?

Mathematics science pertaining to the collection, analysis, interpretation or explanation, and presentation of data

Summarizing information to aid understanding

Drawing conclusions from data

Estimating the present or predicting the future

Random event/process/variableOutcome cannot be predicted Ex: sum of the numbers on two rolled dice Ex: stock market Ex: fluctuations in water pressure in a municipal water supply

http://www.scc.ms.unimelb.edu.au/whatisstatistics/

Page 3: Statistics What is statistics? Where are statistics used?

Uncertainty in Measurements

Where in engineering do we have to worry about uncertainty? Measurements.

Ask 50 people to measure the length of a football field with a meter stick…

Would each person get the same answer? Why or why not?

Field doesn’t change in size; therefore, errors are in the measurements

(misreading tick marks, didn’t measure in a straight line, etc.)

Page 4: Statistics What is statistics? Where are statistics used?

Uncertainty in Measurements

So, how long is the football field?

How confident are we that the length is the average value?

Estimate the length by taking the average.

Calculate the standard deviation about the average.

Page 5: Statistics What is statistics? Where are statistics used?

Mean, Median, and Mode

Mean: the sum of all a list divided by the number of items in the list

1, 2, 4, 5, 5, 8, 9, 12, 15

1, 2, 4, 5, 5, 7, 8, 9, 12, 15

Median: the number separating the higher half of a set of numbers from the lower half

Mode: the value that occurs the most frequently in a set of numbers

Page 6: Statistics What is statistics? Where are statistics used?

When do you use…

Median

Mode

When you know that a distribution is skewed When you believe that a distribution might be skewed When you have a small number of objects Why? To combat the effect of outliers.

When there are many numbers and the frequency of the numbers progress smoothly When you have non-numerical data (categorical data)

Page 7: Statistics What is statistics? Where are statistics used?

Standard Deviation and Variance

Measures of how spread out a distribution is i.e. Measures of variability

Variance: average squared deviation of each number from its mean Summation

Standard Deviation: square root of the variance Most commonly used measure of spread

Page 8: Statistics What is statistics? Where are statistics used?

Definitions

Event/realization Rolling of a pair of dice, taking of a measurement, performing an

experiment

Outcome Result of rolling the dice, taking the measurement, etc.

Deterministic event Event whose outcome can be predicted realization after realization

Ex: measured length of a table to the nearest cm

Probability Estimate of the likelihood that a random event will product a certain

outcome

Page 9: Statistics What is statistics? Where are statistics used?

Probability Distribution Functions

If a random event is repeated many times Produces a distribution of possible outcomes, f(n) The probability distribution function represents the distribution as the

percentage of occurrences of each outcome

Consider rolling a die. Each side is equally likely to land face up:

(Uniform Distribution: f(1) = f(2) = f(3) = f(4) = f(5) = f(6) = 1/6)

Distribution function

Outcome

Page 10: Statistics What is statistics? Where are statistics used?

Probability Distribution Functions

Consider rolling a pair of dice: If outcome of this event is the sum of the dice, what does the distribution look like?

6 x 6 = 36 possible ways for the pair of dice to land.

11 possible outcomes (pairs can sum to 2 12).

For example,

n = 2: (1,1); therefore f(n) = 1/36.

n = 7: (1,6) (2,5) (3,4) (4,3), (5,2), (6,1) ; therefore f(n) = 6/36.

Page 11: Statistics What is statistics? Where are statistics used?

Probability Distribution Functions

Most well-known: bell-shaped distribution

(More formally, Gaussian or normal distribution).

Page 12: Statistics What is statistics? Where are statistics used?

Normal Distribution

Continuous probability distribution Distributions with the same general shape

Symmetric with scores more concentrated in the middle than in the tails

“Bell shaped” Height is determined by the mean and

the standard deviation QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Page 13: Statistics What is statistics? Where are statistics used?

Probability Distribution Functions

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

f(x)

Page 14: Statistics What is statistics? Where are statistics used?

Why are Normal Distributions Important?

1) Many psychological and educational variables are distributed approximately normally- Reading ability, introversion, job satisfaction, etc

2) It’s easy for mathematical statisticians to work with

3) If mean and standard deviations of a normal distribution are known, it is easy to convert back and forth from raw scores to percentiles

Page 15: Statistics What is statistics? Where are statistics used?

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Standard Normal Distribution

Normal distribution with a mean of 0 and a standard deviation of 1.

Normal distributions can be transformed to standard normal distributions by:

This determines how many standard deviations above or below the mean a particular score is.

Ex: Score of 70 on a test with a mean of 50 and standard deviation of 10. Z = 2.

(2 standard deviations above the mean)

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

http://www.oswego.edu/~srp/stats/z.htm

Page 16: Statistics What is statistics? Where are statistics used?

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Page 17: Statistics What is statistics? Where are statistics used?

Percentiles

If the mean and standard deviation of a normal distribution are known, you can find out the percentile rank of a person obtaining a specific score

Ex: Introductory Psychology test normally distributed with a mean of 80 and a standard deviation of 5

What is the percentile rank of a person with a score of 70?

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

The proportion of the area below 70 is equal to the proportion of the scores below 70

z = (70-80)/5 = -2

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

f(x)QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

Page 18: Statistics What is statistics? Where are statistics used?

Percentiles

What about a person scoring 75 on the test?

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

The proportion of the area below 75 is the same as the proportion of scores below 75.

z = (75-80)/5 = -1

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

Page 19: Statistics What is statistics? Where are statistics used?

Percentiles

What about a person scoring 90 on the test?

This graph shows that most people scored below 90.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

z = (90 - 80)/5 = 2

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

Page 20: Statistics What is statistics? Where are statistics used?

Percentiles

The test has a mean of 80 and a standard deviation of 5. What score do you need to be in the 75th percentile?

Page 21: Statistics What is statistics? Where are statistics used?

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Page 22: Statistics What is statistics? Where are statistics used?

Percentiles

The test has a mean of 80 and a standard deviation of 5. What score do you need to be in the 75th percentile?

Look at the z table to find the z associated with 0.75- The value of z is 0.674- This means that you need to be .674 standard deviationsabove the mean

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Standard deviation is 5, so(5)(.674) = 3.37Need to be 3.37 points above the mean

80+3.37 = 83.37Round to get a score of 83

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

Page 23: Statistics What is statistics? Where are statistics used?

Sampling Distribution

Say you compute the mean of a sample of 10 numbers; the value you obtain will not equal the population(entire set) mean exactly

Sampling distribution of the mean is a theoretical distribution that is approached as the number of samples increases

Population with mean of and standard deviation of , the sampling distribution of the mean has mean of and standard deviation (called standard error of the mean):

where n is the sample size

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture. QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Page 24: Statistics What is statistics? Where are statistics used?
Page 25: Statistics What is statistics? Where are statistics used?