week3 1
Changing Unit of Measurement• A linear transformation changes the original value x into a
new variable xnew . • xnew is given by an equation of the form,
• Example 1.21 on page 45 in IPS.(i) A distance x measured in km. can be expressed in
miles as follow, .
(ii) A temperature x measured in degrees Fahrenheit can beconverted to degrees Celsius by
x a b xn ew = +
0 .6 2x xn ew =
5 160 5( 32)9 9 9x x xnew= − =− +
week3 2
Effect of a Linear Transformation
• Multiplying each observation in a data set by a number bmultiplies both the measures of center (mean, median, and trimmed means) by b and the measures of spread (range, standard deviation and IQR) by |b| that is the absolute value of b.
• Adding the same number a to each observation in a data set adds a to measures of center, quartiles and percentiles but does not change the measures of spread.
• Linear transformations do NOT change the overall shapeof a distribution.
week3 3
Measure x xnew
Mean
Median M
Mode
Range R
IQR IQR
Stdev s
a+bM
Mode a+bMode
x xba +
RbIQRb
sb
week3 4
Example 1• A sample of 20 employees of a company was taken and
their salaries were recorded. Suppose each employee receives a $300 raise in the salary for the next year. State whether the following statements are true or false.
a) The IQR of the salaries will i. be unchangedii. increase by $300iii. be multiplied by $300
b) The mean of the salaries will i. be unchangedii. increase by $300iii. be multiplied by $300
week3 5
Density curves
• Using software, clever algorithms can describe a distribution in a way that is not feasible by hand, by fitting a smooth curveto the data in addition to or instead of a histogram. The curvesused are called density curves.
• It is easier to work with a smooth curve, because histogram depends on the choice of classes.
• Density CurveDensity curve is a curve that
is always on or above the horizontal axis.has area exactly 1 underneath it.
• A density curve describes the overall pattern of a distribution.
week3 6
• The area under the curve and above any range of values is the relative frequency (proportion) of all observations that fall in that range of values.
• Example: The curve below shows the density curve for scores in an exam and the area of the shaded region is the proportion of students who scores between 60 and 80.
week3 7
Median and mean of Density Curve
• The median of a distribution described by a density curve is the point that divides the area under the curve in half.
• A mode of a distribution described by a density curve is a peak point of the curve, the location where the curve is highest.
• Quartiles of a distribution can be roughly located by dividing the area under the curve into quarters as accurately as possible by eye.
week3 8
Normal distributions• An important class of density curves are the symmetric
unimodal bell-shaped curves known as normal curves. They describe normal distributions.
• All normal distributions have the same overall shape.
• The exact density curve for a particular normal distribution is specified by giving its mean μ (mu) and its standard deviation σ (sigma).
• The mean is located at the center of the symmetric curve and is the same as the median and the mode.
• Changing μ without changing σ moves the normal curve along the horizontal axis without changing its spread.
week3 10
• There are other symmetric bell-shaped density curves that are not normal e.g. t distribution.
• Normal density function is mathematical model of process producing data.
• If histogram with bars matching normal density curve, data is said to have a normal distribution.
• Notation: A normal distribution with mean μ and standard deviation σ is denoted by N(μ, σ).
week3 11
The 68-95-99.7 ruleIn the normal distribution with mean μ and standard deviation σ ,
Approx. 68% of the observations fall within σ of the mean μ.Approx. 95% of the observations fall within 2σ of the mean μ.Approx. 99.7% of the observations fall within 3σ of the mean μ.
week3 12
Example 1.23 on p72 in IPS• The distribution of heights of women aged 18-24 is
approximately N(64.5, 2.5), that is ,normal with mean μ = 64.5 inches and standard deviation σ = 2.5 inches.
• The 68-95-99.7 rule says that the middle 95% (approx.) of women are between 64.5-5 to 64.5+5 inches tall.The other 5% have heights outside the range from 59.5 to 69.5 inches, and 2.5% of the women are taller than 69.5 .
• Exercise:1) The middle 68% (approx.) of women are between ____to ___
inches tall.2) ___% of the women are taller than 66.75. 3) ___% of the women are taller than 72.
week3 13
Standardizing and z-scores
• If x is an observation from a distribution that has mean μ and standard deviation σ , the standardized value ofx is given by
• A standardized value is often called a z-score.
• A z-score tells us how many standard deviations the original observation falls away from the mean of the distribution.
• Standardizing is a linear transformation that transform the datainto the standard scale of z-scores. Therefore, standardizing does not change the shape of a distribution, but changes the value ofthe mean and stdev.
xz μσ−=
week3 14
Example 1.26 on p61 in IPS• The heights of women is approximately normal with mean
μ = 64.5 inches and standard deviation σ = 2.5 inches.
• The standardized height is
• The standardized value (z-score) of height 68 inches is
or 1.4 std. dev. above the mean.
• A woman 60 inches tall has standardized height
or 1.8 std. dev. below the mean.
6 4 .52 .5
h eig h tz −=
6 8 6 4 .5 1 .42 .5z −= =
60 64.5 1.82.5z −= =−
week3 15
The Standard Normal distribution• The standard normal distribution is the normal distribution
N(0, 1) that is, the mean μ = 0 and the sdev σ = 1 .• If a random variable X has normal distribution N(μ, σ), then the
standardized variable
has the standard normal distribution.
• Areas under a normal curve represent proportion of observations from that normal distribution.
• There is no formula to calculate areas under a normal curve. Calculations use either software or a table of areas. The table and most software calculate one kind of area: cumulative proportions . A cumulative proportion is the proportion of observations in a distribution that fall at or below a given value and is also the area under the curve to the left of a given value.
XZ μσ−=
week3 16
The standard normal tables• Table A gives cumulative proportions for the standard
normal distribution. The table entry for each value z is the area under the curve to the left of z, the notation used is P( Z ≤ z).
e.g. P( Z ≤ 1.4 ) = 0.9192
17
Standard Normal Distributionz .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.02.12.22.32.42.52.62.72.82.93.0
.5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
.5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
.5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
.6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
.6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
.6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
.7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
.7580 .7611 .7642 .7673 .7703 .7734 .7764 .7794 .7823 .7852
.7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
.8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
.8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
.8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
.8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
.9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
.9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
.9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
.9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
.9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
.9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
.9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
.9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
.9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
.9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
.9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
.9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
.9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
.9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
.9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974
.9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
.9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986
.9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990
The table shows area to left of ‘z’ under standard normal curve
week3 18
The standard normal tables - Example
• What proportion of the observations of a N(0,1) distribution takes values
a) less than z = 1.4 ?
b) greater than z = 1.4 ?
c) greater than z = -1.96 ?
d) between z = 0.43 and z = 2.15 ?
week3 19
Properties of Normal distribution
• If a random variable Z has a N(0,1) distribution then P(Z = z)=0. The area under the curve below any point is 0.
• The area between any two points a and b (a < b) under the standard normal curve is given by
P(a ≤ Z ≤ b) = P(Z ≤ b) – P(Z ≤ a)
• As mentioned earlier, if a random variable X has a N(μ, σ) distribution, then the standardized variable
has a standard normal distribution and any calculations about Xcan be done using the following rules:
σμ−
=XZ
week3 20
• P(X = k) = 0 for all k.
• The solution to the equation P(X ≤ k) = p isk = μ + σzp
Where zp is the value z from the standard normal table that has area (and cumulative proportion) p below it, i.e. zp is the pth percentile of the standard normal distribution.
( ) ⎟⎠⎞
⎜⎝⎛ −
≤=≤•σμaZPaXP
( ) ⎟⎠⎞
⎜⎝⎛ −
≤−=≥•σμbZPbXP 1
( ) ⎟⎠⎞
⎜⎝⎛ −
≤≤−
=≤≤•σμ
σμ bZaPbXaP
week3 21
Questions1. The marks of STA221 students has N(65, 15) distribution.
Find the proportion of students having marks (a) less then 50. (b) greater than 80.(c) between 50 and 80.
2. Example 1.30 on page 65 in IPS:Scores on SAT verbal test follow approximately the N(505, 110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT?
3. The time it takes to complete a stat220 term test is normally distributed with mean 100 minutes and standard deviation 14 minutes. How much time should be allowed if we wish to ensure that at least 9 out of 10 students (on average) can complete it? (final exam Dec. 2001)
week3 22
4. General Motors of Canada has a deal: ‘an oil filter and lube jobin 25 minutes or the next one free’. Suppose that you worked forGM and knew that the time needed to provide these services was approximately normal with mean 15 minutes and std. dev. 2.5 minutes. How many minutes would you have recommended to put in the ad above if it was decided that about 5 free services for 100 customers was reasonable?
5. In a survey of patients of a rehabilitation hospital the mean length of stay in the hospital was 12 weeks with a std. dev. of 1 week.The distribution was approximately normal.
a) Out of 100 patients how many would you expect to stay longer than 13 weeks?
b) What is the percentile rank of a stay of 11.3 weeks?c) What percentage of patients would you expect to be in longer
than 12 weeks?d) What is the length of stay at the 90th percentile?e) What is the median length of stay?
week3 23
Normal quantile plots and their use• A histogram or stem plot can reveal distinctly nonnormal
features of a distribution.• If the stem-plot or histogram appears roughly symmetric
and unimodal, we use another graph, the normal quantileplot as a better way of judging the adequacy of a normal model.
• Any normal distribution produces a straight line on the plot.
• Use of normal quantile plots:If the points on a normal quantile plot lie close to a straight line, the plot indicates that the data are normal. Systematic deviations from a straight line indicate a nonnormal distribution. Outliers appear as points that are far away from the overall pattern of the plot.
week3 24
460 470 480 490 500 510 520 530 540
0
5
10
15
value
Freq
uenc
y
• Histogram, the nscores plot and the normal quantile plot for data generated from a normal distribution (N(500, 20)).
450 500 550
1
5
10
20304050607080
90
95
99
Data
Per
cent
Normal Probability Plot for value
ML Estimates
Mean:
StDev:
500.343
17.4618
week3 25
0 5 10
0
5
10
value
Freq
uenc
y
• Histogram, the nscores plots and the normal quantile plot for data generated from a right skewed distribution
week3 26
0 5 10
1
5
10
20
3040506070
80
90
95
99
D ata
Per
cent
Norm al P robability P lot for value
M L Estim ates
M ean:
StD ev:
2.64938
2.17848
week3 27
0 .25 0 .35 0 .45 0 .55 0 .65 0 .75 0 .85 0 .95 1 .05
0
5
10
value
Freq
uenc
y
• Histogram, the nscores plots and the normal quantile plot for data generated from a left skewed distribution
week3 28
0.50 0.75 1.00 1.25
1
5
10
20
3040506070
80
90
95
99
Data
Per
cent
Normal Probability Plot for value
ML Estimates
Mean:
StDev:
0.8102
0.161648
week3 29
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0
1
2
3
4
5
6
7
8
9
value
Freq
uenc
y
• Histogram, the nscores plots and the normal quantile plot for data generated from a uniform distribution (0,5)
week3 30
-2 -1 0 1 2 3 4 5 6
1
5
10
20
3040506070
80
90
95
99
D ata
Per
cent
Norm al Probability P lot for value
M L Estim ates
M ean:
StD ev:
2.21603
1.46678
week3 31
Looking at data - relationships
• Two variables measured on the same individuals are associatedif some values of one variable tend to occur more often with some values of the second variable than with other values of that variable.
• When examining the relationship between two or more variables, we should first think about the following questions:– What individuals do the data describe?– What variables are present? How are they measured?– Which variables are quantitative and which are categorical? – Is the purpose of the study is simply to explore the nature of
the relationship, or do we hope to show that one variable can explain variation in the other?
week3 32
Response and explanatory variables
• A response variable measure an outcome of a study. An explanatory variable explains or causes changes in the response variables.
• Explanatory variables are often called independent variables and response variables are called dependent variables. The ides behind this is that response variables depend on explanatory variables.
• We usually call the explanatory variable x and the response variable y.
week3 33
Scatterplot• A scatterplot shows the relationship between two
quantitative variables measured on the same individuals.• Each individual in the data appears as a point in the plot
fixed by the values of both variables for that individual.• Always plot the explanatory variable, if there is one, on the
horizontal axis (the x axis) of a scatterplot.• Examining and interpreting Scatterplots
– Look for overall pattern and striking deviations from that pattern.
– The overall pattern of a scatterplot can be described by the form, direction and strength of the relationship.
– An important kind of deviation is an outlier, an individual value that falls outside the overall pattern.
week3 34
Example
• There is some evidence that drinking moderate amounts of wine helps prevent heart attack. A data set contain information on yearly wine consumption (litters per person) and yearly deaths from heart disease (deaths per 100,000 people) in 19 developed nations. Answer the following questions.
• What is the explanatory variable?• What is the response variable?• Examine the scatterplot below.
week3 36
• Interpretation of the scatterplot– The pattern is fairly linear with a negative slope. No outliers.– The direction of the association is negative . This means that
higher levels of wine consumption are associated with lower death rates.
– This does not mean there is a causal effect. There could be lurking variables. For example, higher wine consumption could be linked to higher income, which would allow better medical care.
• MINITAB command for scatterplot
Graph > Plot
week3 37
Categorical variables in scatterplots• To add a categorical variable to a scatterplot, use a different
colour or symbol for each category.• The scatterplot below shows the relationship between the
world record times for 10,000m run and the year for both men and women.
F M
200019501900
2300
2200
2100
2000
1900
1800
1700
1600
Year
Tim
e (s
econ
ds)
week3 38
Correlation• A sctterplot displays the form, direction and strength of the
relationship between two quantitative variables.• Correlation (denoted by r) measures the direction and
strength of the liner relationship between two quantitative variables.
• Suppose that we have data on variables x and y for n individuals. The correlation r between x and y is given by
( )
yx
n
iiin
i y
i
x
i
ss
yxnyx
nsyy
sxx
nr
−⎥⎦
⎤⎢⎣
⎡
−=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛ −⎟⎟⎠
⎞⎜⎜⎝
⎛ −−
=∑
∑ =
=
1
1 11
11
week3 39
Example• Family income and annual savings in thousand of $ for a sample
of eight families are given below. savings income C3 C4 C5
1 36 -1.42887 -1.45101 2.073312 39 -1.02062 -1.03144 1.05271
2 42 -0.61237 -0.61187 0.374695 45 -0.20412 -0.19230 0.039255 48 0.20412 0.22727 0.04433
6 51 0.61237 0.64684 0.396117 54 1.02062 1.06641 1.088408 56 1.42887 1.34612 1.92343
Sum of C5 = 6.99429
• r = 6.99429/7 = 0.999185• MINITAB command: Stat > Basic Statistics > Correlation
week3 40
Properties of correlation• Correlation requires both variables to be quantitative and make no
use of the distinction between explanatory and response variables.• Correlation r has no unit if measurement.• Positive r indicates positive association between the variables and
negative r indicates negative association.• Correlation measures the strength of only the linear relationship
between two variables, it does not describe curved relationship!• r is always a number between –1 and 1.
Values of r near 0 indicates a weak linear relationship.The strength of the linear relationship increases as r moves away from 0. Values of r close to –1 or 1 indicates that thepoints lie close to a straight line.r is not resistant. r is strongly affected by a few outliers.
week3 42
Question from Term test, summer 99• MINITAB analyses of math and verbal SAT scores is given below.
Variable N Mean Median TrMean StDev SE MeanVerbal 200 595.65 586.00 595.57 73.21 5.18Math 200 649.53 649.00 650.37 66.35 4.69GPA 200 2.6300 2.6000 2.6439 0.5803 0.0410
Variable Minimum Maximum Verbal 361.00 780.00 Math 441.00 800.00 GPA 0.3000 3.9000
Stem-and-leaf of Verbal N = 200Leaf Unit = 10
1 3 64 4 03419 4 56688888888999952 5 000000122222222333333333444444444
(56) 5 5555555555555666666677777777777777888888888888888999999992 6 0000000001111111122222233333333344444444444444445 6 55555566666666677888888888999915 7 00111122445 7 55568
week3 43
Stem-and-leaf of Math N = 200Leaf Unit = 10
1 4 43 4 7912 5 00122223438 5 55555666677777778888889999(63) 6 00000000000000111111111111222222222222222233333333334444444444499 6 55555555566666666666666777777777778888888999999951 7 00000000001111111111111222222233333444412 7 55667777892 8 00
400 500 600 700 800
0
10
20
30
Math
Freq
uenc
y
400 500 600 700 800
0
10
20
Verbal
Freq
uenc
y
week3 44
a) Find the 25th percentile, 75th percentile and the IQR of the math SAT scores.
b) You were one of the students of this study and your math SAT score was 532. What is your z-score and percentile standing?
c) If the math SAT scores were in fact left (negatively) skewed, but the mean was still 650, what could you say about the percentile standing of someone who obtains a score of 650?
d) What is the class width ?i) of the histogram for verbal SAT scores?ii) of the stemplot of the verbal SAT scores?
e) Describe both the verbal and math score distributions and compare one with the other.
week3 45
g) Give a rough sketch of how a normal probability plot would look if the verbal scores were
i. Right (positively) skewedii. Uniform in shape
h) For verbal scores, aside from running through the data and tallying, can you determine the approx. percentage of scores which fall between 523 and 668? If so give the percentage.