By: David NegrelliΣα
δ2
µ
The ability to describe data in various ways
has always been important. The need to
organize masses of information has led...
 Frequency distribution
 Measures of central tendency
 Measures of dispersion
 Statistical significance
 T-test calcu...
Data is collected and often organized into
formats that are interpreted easily.
Example: Plant height due to the applicati...
Height (cm)
Numberofplants
8 9
9
7 8 9 10 11 12 13 14 15 16
10
10
11
11
12
12
12
12
13
13
14 15
10 14 11 12 15
15 12 13 14...
 Median- The middle number in a set of data.
 Mode- The number within the set of data that
appears the most frequently.
...
 Variance- Determined by averaging the squared
difference of all the values from the mean.
- symbolized by δ2
δ2
= Σ (х –...
 Standard Deviation- Is a measure of
dispersion that defines how an individual
entry differs from the mean.
 - calculate...
 The red area represents the first standard deviant.
 68% of the data falls within this area.
 Calculated by x ± δ
 Th...
 Statistical significance is calculated by
determining:
 if the probability differences between sets of data
occurred by...
 Example of Null Hypothesis:
The mean weight of college football
players is not significantly different
from professional...
 Statistical test that helps to show if there is a
real difference between different treatments
being tested in a control...
( )n1 n2
 Two basic formulas for calculating an uncorrelated t
test.
∙ 1 + 1
x1 – x2
( n1 – 1)δ2
1 + ( n2 – 1) δ2
2
n1 + ...
 Represents the number of independent
observations in a sample.
 Is a measure that states the number of
variables that c...
 Is determined by the researcher.
 Symbolized by α
 Is affected by the sample size and the nature of the
experiment.
 ...
 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 re...
 The following information is needed in a summary
table
Mean
Variance
Standard deviation
1SD (68% Band)
2 SD (95% Band)
3...
 Example: Data obtained from a experiment
comparing the number of un-popped seeds in
popcorn brand A and popcorn brand B....
 Determine mean, variance and standard deviation
of samples.
Mean xA = Σx
n
= 26+22+30+34
4
= 23
= Σx
n
Mean xB
= 32+35+2...
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 ...
popcorn A
Popcorn B
δ= √ δ2Standard deviation:
√ 60 = 7.75
√ 44.67 = 6.68
Finding Calculated t
t = 23 - 30
x1 – x2t =
√ δ2
1 + δ2
2
n
√
60+ 44.67
4
= 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=...
Mean
Variance
Standard deviation
1SD (68% Band)
2 SD (95% Band)
3 SD (99% Band)
Number
Results of t test
Descriptive stati...
 Write a topic sentence stating the independent and
dependent variables and a reference to a table or
graph.
 Write sent...
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  1. 1. By: David NegrelliΣα δ2 µ
  2. 2. 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.
  3. 3.  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
  4. 4. 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
  5. 5. Height (cm) Numberofplants 8 9 9 7 8 9 10 11 12 13 14 15 16 10 10 11 11 12 12 12 12 13 13 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
  6. 6.  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
  7. 7.  Variance- Determined by averaging the squared difference of all the values from the mean. - symbolized by δ2 δ2 = Σ (х – х)2 n-1
  8. 8.  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
  9. 9.  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
  10. 10.  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.
  11. 11.  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
  12. 12.  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.
  13. 13. ( )n1 n2  Two basic formulas for calculating an uncorrelated t test. ∙ 1 + 1 x1 – x2 ( n1 – 1)δ2 1 + ( n2 – 1) δ2 2 n1 + n2 – 2√ t = Unequal sample size Equal sample size x1 – x2t = √ δ2 1 + δ2 2 n
  14. 14.  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)
  15. 15.  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.
  16. 16.  A probability table is used  First determine degrees of freedom  Decide the level of significance
  17. 17.  Example: degrees of freedom= 4 α= .05  The critical value of t= 2.776
  18. 18.  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.
  19. 19.  The following information is needed in a summary table Mean Variance Standard deviation 1SD (68% Band) 2 SD (95% Band) 3 SD (99% Band) Number Results of t test Descriptive statistics
  20. 20.  Example: Data obtained from a experiment comparing the number of un-popped seeds in popcorn brand A and popcorn brand B. A B 26 32 22 35 30 20 34 33 Is the difference significant?
  21. 21.  Determine mean, variance and standard deviation of samples. Mean xA = Σx n = 26+22+30+34 4 = 23 = Σx n Mean xB = 32+35+20+33 4 = 30
  22. 22. 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
  23. 23. popcorn A Popcorn B δ= √ δ2Standard deviation: √ 60 = 7.75 √ 44.67 = 6.68
  24. 24. Finding Calculated t t = 23 - 30 x1 – x2t = √ δ2 1 + δ2 2 n √ 60+ 44.67 4 = 7 √26.17 = 7 5.12 = 1.38
  25. 25. 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.
  26. 26. 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 30 60 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
  27. 27.  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.
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