Statistical Analysis

Definition of Statistics

Descriptive Statistics: numerical facts, figures and information

Goal:1. Describe a set of numbers

2. Make accurate inferences about process/behavior based upon incomplete information

Statistical Steps

Steps1. Gather data2. Organize data3. Analyze data

What is Data

Types of data

Discrete and Continuous DataData can be Discrete or Continuous.Discrete data is counted, Continuous data is measured

Population and Sample

Population: collection of all elements of interest Sample: subset of the population

Population Sample

Population

Sample

Data

Numerical Measures

Measurement of central tendency Mean Median mode

Measurement of variations Range Variance Standard deviation

Arithmetic mean (AM)

The arithmetic mean is the "standard" average, often simply called the "mean".

Mean

Arithmetic mean (AM)

For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is

Mean

The mean is the average value

x x i

n

x i

N

Sample mean

Population mean

Median

The middle value when the numbers are arranged in ascending or descending order.

12345

medianEx:

Mode

The data value that occurs with greatest frequency

11135

modeEx:

Central Tendency Example

Test scores:

x i

N

(98 96 94 ... 64 44)

168298 83

96 8394 8393 8088 7587 7286 6486 44

Median

(86 83)

284.5

Mode = 83

Range

Difference between the largest and smallest value in the dataset

12345

Ex: Range = 5-1 =4

Dispersion

Deviation just means how far from the normal

Root mean square

RMS is a statistical measure of the magnitude of a varying quantity. It is especially useful when variants are positive and negative

Variance

Measurement of the dispersion of values from the mean

Sample variance

Population variance

s2 x i x 2

n 1

2 x i 2

N

Statistics for Business and Economics,5th ed., pg68.

Standard Deviation Definition:

     Standard deviation is a statistical measure of spread or variability. The standard deviation is the root mean square (RMS) deviation of the values from their arithmetic mean.

Standard Deviation

Standard Deviation

The positive square root of the variance

Sample standard deviation

Population standard deviation

s s2

2

Percentile The value such that p% of the total items lie below the value

Lower quartile, middle quartile, upper quartile

Steps1. Arrange data in ascending order2. Compute an index i as follows:

3. If i is not an integer, round up. If i is an integer, the p% is the average between the values in position i and i+1.

i p

100

n

histogram

A Histogram is a graphical display of data using bars of different heights

histogram

It is similar to a Bar Chart, but a histogram groups data into ranges

Frequency & Frequency Distribution

FrequencyFrequency is how often something occurs.

Frequency DistributionBy counting frequencies we can make a Frequency Distribution table.

Probability

P(A) = number of favorable outcomes total number of possible outcomes

Areas of Probability:• simple events• combinations of events

Probability Distribution

• Continuous Distribution: continuous scale

• Discrete Distribution: discrete values

Discrete Vs. Continuous Data

Discrete Examples Yes/No Throws of a die Go / No go Heads/Tails

Continuous Examples segment time degrees of temperature measurements like

inches, miles, feet, etc weight

Continuous Distribution

• Uniform Distribution

• Exponential Distribution

• Normal Distribution

Normal Distribution

Symmetric distribution Highest point occurs at mean Mean, median and mode are at center point

Statistics for Business and Economics,5th ed., pg184.

f (x) 1

2e x 2 / 2 2

Normal Distribution

34.13%

13.06%

2.14%0.13%

34.13%

13.06%

2.14%0.13%

68.26%

99.73%

95.46%

68.26% of the population is within +/- 1 of the

-3

Normal Distribution

Assumed Normality

Z-scores

• A Z score is a data point's position between the mean and another location as measured by the number of standard deviations.

• Z is a universal measurement because it can be applied to any unit of measure.

z x

z 0.000 0.001 0.002 0.003 0.004 0.0050.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.52000.1 0.5398 0.5438 0.5478 0.5518 0.5558 0.55980.2 0.5793 0.5832 0.5871 0.5910 0.5949 0.59880.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.63690.4 0.6554 0.6591 0.6628 0.6665 0.6702 0.67390.5 0.6915 0.6950 0.6985 0.7020 0.7055 0.70900.6 0.7257 0.7291 0.7325 0.7359 0.7393 0.74270.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.77350.8 0.7881 0.7910 0.7939 0.7968 0.7997 0.80260.9 0.8159 0.8186 0.8213 0.8240 0.8267 0.82941.0 0.8413 0.8438 0.8463 0.8488 0.8513 0.8538

Standard Normal Probability Table

Confidence Level

Confidence Level

Standard Normal Distribution

Normal Vs Standard Normal

3210-1-2-3

0.4

0.3

0.2

0.1

0.0

Normal

CDF

Z

Standard Normal Probability Area

0.7393 0.2607

= 0.64

Distribution Mean

Confidence Level and interval

Precision is defined in terms of standard deviation.

= one sigma or 2 or 3 according to the confidence level

Confidence Level and interval

If your instrument measures in "2"s then any value between 7 and 9 is measured as "8"

Confidence Level

Confidence Interval

Confidence level and Interval

Confidence Level

Peak removal according to confidence level

Peak removal according to confidence level

Data Regression

Data Regression

Peak removal

Peak removal