صفحه 1:
Statistics
From BSCS: Interaction of
experiments and ideas, 2°¢ Edition.
Prentice Hall, 1970 and Statistics
for the Utterly Confused by Lloyd
Jaisingh, McGraw-Hill, 2000
صفحه 2:
وگ
is the fraction of the variation in the
values of y that is explained by the
least-squares regression line of y
on x.
Exanple: IP? = 0.000 ta the gropk to the kA, his wee
تاه با 00% oP pae's qrade is acerunited Por by the foe
Cites مق wih otieadsare. Che other 99% could be due
و( اه له و و
Chose Pteuddace
صفحه 3:
What is statistics?
a branch of mathematics that provides
techniques to analyze whether or not
your data is significant (meaningful)
Statistical applications are based on
probability statements
Nothing is “proved” with statistics
Statistics are reported
Statistics report the probability that
similar results would occur if you
repeated the experiment
صفحه 4:
Statistics deals with
numbers
* Need to know nature of numbers collected
— Continuous variables: type of numbers
associated with measuring or weighing; any
value in a continuous interval of
measurement.
* Examples:
— Weight of students, height of plants, time to flowering
— Discrete variables: type of numbers that are
counted or categorical
+ Examples:
— Numbers of boys, girls, insects, plants
صفحه 5:
Can you figure out...
Which type of numbers (discrete or
continuous?)
— Numbers of persons preferring Brand X in 5
different towns
—The weights of high school seniors
—The lengths of oak leaves
—The number of seeds germinating
— 35 tall and 12 dwarf pea plants
— Answers: all are discrete except the 2"? and
3" examples are continuous.
صفحه 6:
Populations and Samples
٠ Population includes all members of a group
— Example: all 98 grade students in America
— Number of 9* grade students at SR
- No absolute number
٠ Sample
— Used to make inferences about large populations
— Samples are a selection of the population
— Example: 6'* period Accelerated Biology
+ Why the need for statistics?
— Statistics are used to describe sample populations as
estimators of the corresponding population
— Many times, finding complete information about a
population is costly and time consuming. We can use
samples to represent a population.
صفحه 7:
Sample Populations
avoiding Bias
Individuals in a sample population
— Must be a fair representation of the
entire pop.
—Therefore sample members must be
randomly selected (to avoid bias)
— Example: if you were looking at strength
in students: picking students from the
football team would NOT be random
صفحه 8:
Is there bias?
A cage has 1000 rats, you pick the first 20 you can
catch for your experiment
A public opinion poll is conducted using the
telephone directory
You are conducting a study of a new diabetes drug;
you advertise for participants in the newspaper and
TV
All are biased: Rats-you grab the slower rats.
Telephone-you call only people with a phone
(wealth?) and people who are listed (responsible?).
Newspaper/TV-you reach only people with
newspaper (wealth/educated?) and TV( wealth?).
صفحه 9:
Statistical Computations (the
Math)
* If you are using a sample population
— Arithmetic Mean (average)
z. =x
N «= {1,2,3,4,5};2=3
The sum of all the score.
divided by the total number of
scores.
— The mean shows that % the members of
the pop fall on either side of an estimated
value: mean
|
صفحه 10:
“Looking at profile of data:
Distribution
* What is the frequency of distribution,
where are the data points?
Distribution Chart of Heights of 100 Control Plants
er of plants in
Class (height of plants: Nu
cm) eat
3 0.0.0.9
10 هر
21 2.0.2.9
30 3.03.9
20 40.4.9
14 5.05.9
6.06.9 2
صفحه 11:
Histogram-Frequency
Distribution Charts
Number of Plants in each Class
sa Number of plants ineach
This is called a “normal” curve or a bell curve
This is an “idealized” curve and is theoretical based on an
infinite number derived from a sample
صفحه 12:
Mode and Median
* Mode: most frequently seen value (if no
numbers repeat then the mode = 0)
* Median: the middle number
—If you have an odd number of data then the
median is the value in the middle of the set
—If you have an even number of data then
the median is the average between the two
middle values in the set.
صفحه 13:
Variance (s2)
* Mathematically expressing the
degree of variation of scores (data)
from the mean
+ A large variance means that the
individual scores (data) of the
sample deviate a lot from the mean.
« Asmall variance indicates the scores
(data) deviate little from the mean
صفحه 14:
مس سار و و سس ولج(
بصخم eu of X = sore, = 2
ween, = td of scores or usher =[ لل 22 _ و
N
OR use the OBR Pucrtivs is Bare
Worksheet for Calculating the
Variance for 7 scores For this problem the population variance is 0.57
XX (X- Wf
5 ۲ 1
3 1 1
4 م6 0
4 0 0
3 1 1
4 0 0
S| 1
bg 4 hip Iunnw xecokte eduluxesrled DC rdovardevs. hice
صفحه 15:
منم Por a Bised GBOPLE جومم سا مطولطله)
anv of; X = sore, uch, =
totd oF scores or uches-(1 = 4 و
vy
Sa L(x 0 xX)
7۱-1 (often.read as “x bar”) is the mean (average value of
Worksheet for Calculating the
Qote the نون موی ts haryer...why?
‘Variance for 7 scores
لي 1
وو كر ge EON) قر ع عر م اع
7-1 7-1 د
3 1 1
۱ For this problem the population variance is 0.57
3 1 1
4 6
S| 1
28 4
جما .وجل صا ©( ©لج لمموجسب ادلب جاداصمج. بصنم | hip
صفحه 16:
ights in Centimeters of Five Randomly Selected Pea Plants Grown at 8-1
Pla Height Deviations Squares of
nt (cm) from mean deviation
from mean
(x) (x; x) (x,- x)?
A 10 2 4
B 7 1 1
© 6 2 4
D 8 0 0
E 9g 1 1
2 ۲ < = (x; x) =0 = (x- x)? = 10
40
X, = score of value; X (bar) = mean; = = sum of
صفحه 17:
عمنو() سا مطلطل() ا۳)
“Dhere were Pave phate; 09; therePore o-
42
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ge LX)
99 10/6 :6 7-1
der با بط ام و همطل سا سس واه وا helps ه00
مت روم عامومجو با مات ای امه ات با
صفحه 18:
Standard Deviation
An important statistic that is also used to
measure variation in biased samples.
S is the symbol for standard deviation
Calculated by taking the square root of
the variance
So from the previous example of pea
plants:
The square root of 2.5 ; s=1.6
Which means the measurements vary
plus or minus +/- 1.6 cm from the mean
صفحه 19:
What does “S” mean?
* We can predict the probability of
finding a pea plant at a predicted
height... the probability of finding a
pea plant above 12.8 cm or below
3.2 cm is less than 1%
* Sis a valuable tool because it reveals
predicted limits of finding a particular
value
صفحه 20:
Pra Plact Oerwal Disttbuticd Curve uth Grt Dev:
صفحه 21:
The Normal Curve and
Standard Deviationer ian:
Cock veriod ke io
اه اه سس
cae
00% of uches Pal
wets +1 or Cl of the
sees
98% oF uches Pal
wikis +O 8, © صب
لس اه رام
(299%) Pal whic
© لد dev write
Same as others
Probably less
than others
Definitely less
than others
wae) ome trae tom
Probably more
than others
Definitely more
than others
2 ۲ 4 ۲
رس امسر راو حلله مد اهنا
صفحه 22:
Standard Error of the Sample
Means
KA Standard Error
A
The mean, the variance, and the std dev
help estimate characteristics of the
population from a single sample
So if many samples were taken then the
means of the samples would also form a
normal distribution curve that would be
close to the whole population.
The larger the samples the closer the
means would be to the actual value
But that would most likely be impossible to
obtain so use a simple method to compute
the means of all the samples
صفحه 23:
A Simple Method for estimating
standard error
5
سب < بر
fn
(Gtoodard error te the colculited stocdard deviatiza divided by the square root oF the Or
اه ام te popubatios
(Gtoodard error و عم بل چاه used to test the retobiiiy oP the cot
.سم 1 here ore (0 core phan ولج و لس deviation خا 0.2
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0.009 represeds vor std dev too scp oF 40 ام
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erase whee we tohe horwer suxopkes, our واه بو سوه وتو to the
fru weoo ude of the popukiica. Dhus, the dstrbuica of the مروت wero
would be tess spread cut ced woud hove a luwer stoodacd deviation.
صفحه 24:
Probability Tests
What to do when you are comparing two
samples to each other and you want to know if
there is a significant difference between both
sample populations
(example the control and the experimental
setup)
How do you know there is a difference
How large is a “difference”?
How do you know the “difference” was caused
by a treatment and not due to “normal”
sampling variation or sampling bias?
صفحه 25:
Laws of Probability
The results of one trial of a chance event do not affect
the results of later trials of the same event. p= 0.5
(a coin always has a 50:50 chance of coming up
heads)
The chance that two or more independent events will
occur together is the product of their changes of
occurring separately. (one outcome has nothing to do
with the other)
Example: What’s the likelihood of a 3 coming up ona
dice: six sides to a dice: p = 1/6
Roll two dice with 3’s p = 1/6 *1/6= 1/36 which means
there’s a 35/36 chance of rolling something else...
Note probabilities must equal 1.0
صفحه 26:
Laws of Probability
(continued)
The probability that either of two or more
mutually exclusive events will occur is the
sum of their probabilities (only one can
happen at a time).
Example: What is the probability of rolling a
total of either 2 or 12?
Probability of rolling a 2 means a 1 on each
of the dice; therefore p = 1/6*1/6 = 1/36
Probability of rolling a 12 means a6 anda
6 on each of the dice; therefore p = 1/36
So the likelihood of rolling either is
1/36+1/36 = 2/36 or 1/18
۰
۰
۰
صفحه 27:
The Use of the Null
Hypothesis
Is the difference in two sample populations
due to chance or a real statistical
difference?
The null hypothesis assumes that there will
be no “difference” or no “change” or no
“effect” of the experimental treatment.
If treatment A is no better than treatment B
then the null hypothesis is supported.
If there is a significant difference between A
and B then the null hypothesis is rejected...
۰
۰
صفحه 28:
T-test or Chi Square? Testing
the validity of the null
hypothesis
Use the T-test (also called Student’s T-
test) if using continuous variables from
a normally distributed sample
populations (ex. Height)
Use the Chi Square (X?) if using discrete
variables (if you are evaluating the
differences between experimental data
and expected or hypothetical data)...
Example: genetics experiments,
expected distribution of organisms.
صفحه 29:
T-test
٠ T-test determines the probability that
the null hypothesis concerning the
means of two small samples is correct
* The probability that two samples are
representative of a single population
(supporting null hypothesis) OR two
different populations (rejecting null
hypothesis)
صفحه 30:
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صفحه 31:
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صفحه 32:
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صفحه 33:
Ose Hest to detervice whether or ont sap popubiog (Boe ( ی Prow the
سلجم اممو ةلال عم جمد
عصد مد / <<
xl (bor x) = wear of Bj xG (bor x) = wear of ©
axl = std error of @; ox@ = std error of
02: :اوه Gaople ® wear =O
Gaople ® wea 29
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(6-9/0 = © std devictiog ucts
صفحه 34:
Comparison of A and B
96 وی 0's wen tes
0 ا اس موس
purve oP poputatiza
Repent Dull Wypotests
صفحه 35:
Octor calruktiors:
coe ماوت هولج واه وولو نها
مه Por pow... ond a box plot وی
hipt/huww.ropkrad col quichodcsltest( cP
صفحه 36:
ات و لول با موه وله مه وت با ارت با وا تاه ۳
وا[
لد لم ۹۰( عع سيق
۲۲ 0 1
Mean sample ۹ N= # in sample | < را
Xy= mean Sample Q N= in Sample 2
Sample | 5
variance of Sample 2 = >33
If samples are equal in Size see nak eal
t= XX |
y= +S
7
صفحه 37:
nt of O, Used by Germinating Seeds of Corn and Pea Plants
ml OJhour | at 25°C
Reading | Com Pea ALow to de his hin BXOCL
Number
1 0.20 0.25
2 0.24 0.23
3 0.22 oat
4 0.21 0.27
5 0.25 0.23
6 0.24 3
7 0.23 0.25
8 0.20 0.28
9 0.21 0.25
10 0.20 0.30
Total 2.20 2.70
Mean 0.22 0.27
Variance | 0028 0106
(Cxxcet Pie tocated tc (Bor Bio Pile Potter
صفحه 38:
the chat che (he pelo مود جاح سل ١ بط ۵ ام للم = زرا
0
اوه هجو مه سا eu of بط روا سل
aroun. اه بط امه وج مت he ما سل
] 2
Albus Dumbledough vs. Pat Stat
| [See
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صفحه 39:
۲
Por (OD decrees of
مس )60-4(
و و سل وه بل
pour tuche toe 2.860
٩۱۳ ای مر ۱ chur te
betwee
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مه 6 606(
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1 ieble oP ches (8% = 0.06(
degrees significance level
of
freedom 20% 10% 5% 2% 1% 01%
1 3078 6314 31821 63-657 636-619
2 1886 20 6965 9925 8
3 1638 3 4541 5841 12.941
4 ۰ 7 ۵ 3.747 4604 ۰ 0
و 1476 015 2.365 4092 9
6 14401943 مرو 370 5-959
7 5 5 2998 3-499 5405
8 1397 ۵ 2896 3355 504
9 1383 33 2.821 3250 47
۱0 372 ۸ 2764 3169 ۰ 87
1 1363. 1-796 2718 3106 4-437
120 1356 2 2681 3055 . 8
13 ۰. 350 171 2650 3012 4221
14 1346 ۱ 2624 ۰ 2977 ۰ 0
15 . 1 ۶3 2602 2947 4.073
16 1337 1-746 2583 2921 4015
17 تا" 9 2567 2898 ۰ 65
18 330 1-734 2552 2878 ۰.22
19 1 1-729 2539 2864 3
2 13299 25 2528 2846 0
2 1-721 2831 3819
2 1717 2819 ۶2
23 na 2807 ۰.7
24 11۱ 2797 ۰ 45
25 108 2.787 ۰ 25
36 1706 719 = 707
صفحه 40:
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Dhe vortrare (5°)oP the core is 0. DOOM ced the peur te DOOAIPO.
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t- xX وا(
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3533032
صفحه 41:
he “2” test
seed P pour تسه اجه ولج thats OD
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veraore
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صفحه 42:
“D table (sucople table with D probubilties
0 20/2[ (لنما مهم) 20 ۰
tails)
0.1 8 1.64
0.05 5 1.96
0.01 33 2.576
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ww dPPereue bet the a OF th ic or the expericectal hypriests (\
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dPPereue
صفحه 43:
Example z-test
* You are looking at two methods of learning
geometry proofs, one teacher uses
method 1, the other teacher uses method
2, they use a test to compare success.
* Teacher 1; has 75 students; mean =85; stdev=3
* Teacher 2: has 60 students; mean =83; stdev=
2 - و2 = (85-83)/V3*2/75 +
22 <
5
ات
= 2/0.4321 = 4.629
صفحه 44:
Example continued
0 را
Detad (ts ont beter thon wetud 2 با ات جوا با
ALO = cherentie Wyzokeoty wink! be teat Drtheal (ts beter دجا sober ©
Whe bw ove kid 2 tot (ower the «nll hypokeoty doe pride fra here wd be wo WPPereccr)
Gp Por the probably oP 0.06 (O% onnPexnee or 08% vexP ecw) tht Deherd ove tort
beter tho لس © ... hat ohort ke = Za 1.645
So 4.629 is greater than the 1.645 (the null hypothesis states that method
1 would not be better and the value had to be less than 1.645; it is not less
therefore reject the null hypothesis and indeed method 1 is better
table (secple table wits O probabstics)
3 Za (one tail) |Za/2 (two
tails)
0.1 8 1.64
0.05 5 1.96
0.01 233 2.576
صفحه 45:
Chi square
* Used with discrete values
* Phenotypes, choice chambers, etc.
* Not used with continuous variables
(like height... use t-test for samples
less than 30 and z-test for samples
greater than 30) 2
* O= observed valu y2 aye
+ E= expected valu 6
مدا حمسي 5 )داه أعصمصام 5) | مجه ومجسمجوط. نمي hep:
صفحه 46:
http://course1.winona.edu/sberg/Equation/chi-
squ2.gif
Observed individuals — Expected individuals
with a given phenotype with a given phenotype
Greek me
2
0-6(
2
ayes
e
Summation => add together a term
for each condition
صفحه 47:
Interpreting a chi square
Calculate degrees of freedom
# of events, trials, phenotypes -1
Example 2 phenotypes-1 =1
Generally use the column labeled 0.05 (which
means there is a 95% chance that any difference
between what you expected and what you
observed is within accepted random chance.
Any value calculated that is larger means you
reject your null hypothesis and there is a
difference between observed and expect values.
صفحه 48:
How to use a chi Square
0.05
3.84
5.99
182
9.49
11.07
12.59
14.07
15.51
16.92
18.81
0.01 0.001
6.64 10.83
921 13.82
11.84 16.27
18.28 18.47
15.09 20.52
16.81 22.46
18.48 24,32
20.09 26.12
21.67 27.88
23.21 29.59
Significant
0.10
2.71
4.60
6.25
18
9.24
10.64
12.02
13.36
14.68
15.99
chart
Probability
0.20
1.64
3.22
4.64
5.99
729
8.56
9.80
11.03
12.24
13.44
0.70 0.50 0.30
1.07
241
3.66
4.88
6.06
1,23
8.38
9.52
10,66
11.78
Nonsignificant
“080
0.90
0.95
Degrees of
Freedom
اسان نله اجه زا جه "|| /نجاها.
صفحه 49:
Spare ۳ دا
Me a Expected ۳5
Exanple 1 )۱۶-۵۵(< )28< :
boo
2, Bs BS, 225.
= 8+ 4 ‘Goo G00 "40
a 0.31540. S097
yas feedom = N-| = &suyo~
