Statistical ppt

By: David Negrelli Σα

δ 2

µ

The ability to describe data in various ways has always been important. The need to organize masses of information has led to the development of formalized ways of describing data. The purpose of this presentation is to introduce the reader to basic tenants of statistics.

Frequency distribution

Measures of central tendency

Measures of dispersion

Statistical significance

T-test calculations

Degrees of freedom

Levels of significance

Finding the critical value of T

Filling out summary table

Writing the results

Data is collected and often organized into formats that are interpreted easily.

Example: Plant height due to the application of fertilizers. Height is given in centimeters (cm.)

10 14 11 12 15

15 12 13 14 13 12 8 12 9 10 13 11 12 8 10 9 16 7 11 9

Height (cm)

Num

ber

of p

lant

s

8 99

7 8 9 10 11 12 13 14 15 16

1010

1111

12

121212

1313

14 15

10 14 11 12 15 15 12 13 14 13 12 8 12 9 10 13 11 12 8 10 9 16 7 11 9

Median- The middle number in a set of data. Mode- The number within the set of data that

appears the most frequently. Mean- The average

a. Denoted by х

b. Calculated by the following

formula Х = Σx

n

Variance- Determined by averaging the squared difference of all the values from the mean.

- symbolized by δ2

δ2 = Σ (х – х)2

n-1

Standard Deviation- Is a measure of dispersion that defines how an individual entry differs from the mean.

- calculated by finding the square root of the

variance. Defines the shape of the normal distribution

curve

δ = √ δ2

The red area represents the first standard deviant. 68% of the data falls within this area. Calculated by x ± δ The green area represents the second standard deviant. 95% of the data falls within the green PLUS the red area. Calculated by x ± 2δ The blue area represents the third standard deviant. 99% of the data falls within blue PLUS the green PLUS the red area. Calculated by x ± 3δ

Statistical significance is calculated by determining: if the probability differences between sets of data

occurred by chance or were the result of the experimental treatment.

Two hypotheses need to be formed: Research hypothesis- the one being tested by the

researcher. Null hypothesis- the one that assumes that any

differences within the set of data is due to chance and is not significant.

Example of Null Hypothesis: The mean weight of college football players is not significantly different from professional football players.

µcf = µpf

µ, ‘mu’ symbol for

Null Hypothesis

Statistical test that helps to show if there is a real difference between different treatments being tested in a controlled scientific trial.

The Student t test is used to determine if the two sets of data from a sample are really different? The uncorrelated t test is used when no

relationship exist between measurements in the two groups.

( )n1 n2

Two basic formulas for calculating an uncorrelated t test.

∙ 1 + 1

x1 – x2

( n1 – 1)δ21 + ( n2 – 1) δ2

2

n1 + n2 – 2√t =

Unequal sample size

Equal sample size

x1 – x2 t =

√ δ21 + δ2

2

n

Represents the number of independent observations in a sample.

Is a measure that states the number of variables that can change within a statistical test.

Calculated by n-1 ( sample size – 1)

Is determined by the researcher. Symbolized by α Is affected by the sample size and the nature of the

experiment. Common levels of significance are

.05, .01, .001 Indicates probability that the researcher made an

error in rejecting the null hypothesis.

A probability table is used First determine degrees of freedom Decide the level of significance

Example: degrees of freedom= 4

α= .05

The critical value of t= 2.776

If the calculated value of t is less than the critical value of t obtained from the table, the null hypothesis is not rejected.

If the calculated value of t is greater than the critical value of t from the table, the null hypothesis is rejected.

The following information is needed in a summary table

MeanVariance

Standard deviation1SD (68% Band)2 SD (95% Band)3 SD (99% Band)

Number

Results of t test

Descriptive statistics

Example: Data obtained from a experiment comparing the number of un-popped seeds in popcorn brand A and popcorn brand B.

A B26 3222 3530 2034 33

Is the difference significant?

Determine mean, variance and standard deviation of samples. Mean xA = Σx

n= 26+22+30+34

4= 23

= Σx

nMean xB = 32+35+20+33

4= 30

variance δ2= Σ (х – х)2

n-1

Popcorn A = ( 26-23)2 + (22-23)2 + (30-23)2 + (34-23)2

3= 9 + 1 + 49 + 121

3= 60

Popcorn B = ( 30-30)2+ (35-30)2 + (20- 30)2 + (33- 30)2

3= 0 + 25 + 100 + 9

3= 44.67

popcorn A

Popcorn B

δ= √ δ2Standard deviation:

√ 60 = 7.75

√ 44.67 = 6.68

Finding Calculated t

t = 23 - 30

x1 – x2 t =

√ δ21 + δ2

2

n

√ 60+ 44.674

= 7

√ 26.17

= 7

5.12 = 1.38

Determine critical value of t• Select level of significance α=.01

• Determine degrees of freedom

degrees of freedom of A= 3

degrees of freedom of B= 3

total degrees of freedom = 6• Critical value of t = 3.707

Calculated value of t =1.38 is less than critical value of t from the table, 3.707.

The null hypothesis is not rejected.

Mean

Variance

Standard deviation

1SD (68% Band)

2 SD (95% Band)

3 SD (99% Band)

Number

Results of t test

Descriptive statistics popcorn A popcorn B

23 3060 44.67

7.75 6.68

15.25 - 30.75 23.32- 36.68

7.50-38.50 16.64-43.36-.25 - 46.25 9.96-50.04

4 4

t= 1.38 df=6 t of 1.38 < 3.707 α=.01

Write a topic sentence stating the independent and dependent variables and a reference to a table or graph.

Write sentences comparing the measures of central tendency of the groups.

Write sentences describing the statistical tests, levels of significance, and the null hypothesis.

Write sentences comparing the calculated value with the required statistical value. Make a statement about rejection of the null hypothesis.

Write a sentence stating support of the research hypothesis by the data.