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Lecture Material Stat PPT
Citation preview
Types of data
Gender male/female
continuousAge in years
ContinuousTime is on a continuous scaleWeight nearest Kg
continuousWeekly expenditure
Continuous
Again a continuous scale but you might argue that it is discreteNumber of siblings
discreteParty Labour, Cons, Lib Dem, other
categoricalTerm accommodation Halls, Home, Private rented, other
categoricalAssignment grade A, B, C, D, E
Ranked because we put grade A as better than grade B better than C etc
Tabulation of data
MalesFemalesTotalHalls75100175Home303868Private rented21930Other437Total130150280
X 360
Pie chart angle formula
Halls 75/130 x360 =208Home 30/130x360= 83Private 21/130x360=58Other 4/130x360= 11
Must add up to 360Pie chart angle calculations
Pie Chart
Chart3
75
30
21
4
males
Sheet1
malesfemales
Halls75100
Home3038
Private rented219
other43
Sheet1
males
Sheet2
females
Sheet3
Presentation of informationA simple bar chart has non touching bars with the height of each bar proportional to the frequency. In Excel a bar chart with vertical bars is called a Column chartAnd with horizontal bars is called a Bar chartMultiple Bar Chart: the bars are split into several to show another variable. In Excel such a bar chart is known as a clustered column with vertical bars and a clustered bar with horizontal bars.Component Bar Chart:components are stacked together to show another variable. In Excel such a bar chart is known as a stacked column with vertical bars and a stacked bar chart with horizontal bars.Percentage Component Bar charts convey the information in percentage form rather than the actual frequency values and thus highlight differences in proportions of one variable.
Frequency table
TimeFrequency BoundariesClass sizeClass midpoint0-
Open ended classesUnder 5 becomes 0 to under 5
40 and over becomes 40 to under 45
Histogramof delivery times
Chart1
0
3
3
7
7
10
10
12
12
8
8
6
6
2
2
2
2
0
Delivery times (days)
Frequency
Histogram of delivery times
histogram1
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0
time (days)
frequency
histogram of delivery times
histogram blank grid
00
50
100
150
200
250
300
350
400
450
500
histogram blank grid
0
0
0
0
0
0
0
0
0
0
0
blank grids
50
100
150
200
250
300
350
400
450
500
blank grids
0
0
0
0
0
0
0
0
0
0
Sheet1
0
0
0
0
0
0
0
0
0
0
histograms2
Time (days)Frequency50
5053
103103
157107
2010157
25121510
3082010
3562012
4022512
452258
308
306
356
352
402
402
452
450
histograms2
Delivery times (days)
Frequency
Histogram of delivery times
cumultive frequencytable
cumultive frequencytable
0
0
0
0
0
0
0
0
0
0
cumultive f graph
0
0
0
0
0
0
0
0
0
0
cumultive f graph measures
uppper boundarycumulative frequency
delivery timefrequencyless than 550
5 -< 103less than 10103
10 -
Frequency polygon of delivery times
Chart1
0
3
7
10
12
8
6
2
2
0
frequency
Delivery times (days)
frequency
frequency polygon
histogram equal
50
53Histogram equal classes
103
107
157
1510
2010
2012
2512
258
308
306
356
352
402
402
452
450
histogram equal
Delivery times (days)
Frequency
Histogram of delivery times
histogram1
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0
time (days)
frequency
histogram of delivery times
unequal classes
50
100
150
200
250
300
350
400
450
500
01.5
51.5
101.5
107
157
1510
2010
2012
2512
258
308
303.3
353.3
403.3
453.3
450
00.3
50.3
100.3
101.4
151.4
152
202
202.4
252.4
251.6
301.6
300.7
350.7
400.7
450.7
450
unequal classes
0
0
0
0
0
0
0
0
0
0
histograms2
0
0
0
0
0
0
0
0
0
0
cumultive frequencytable
1.5
1.5
1.5
7
7
10
10
12
12
8
8
3.3
3.3
3.3
3.3
0
delivery times (days)
frequency per class size 5
histogram using standard class size 5
cumultive f graph
0.3
0.3
0.3
1.4
1.4
2
2
2.4
2.4
1.6
1.6
0.7
0.7
0.7
0.7
0
delivery times (days)
frequency density
histogram using frequency density
frequecny polygon
frequecny polygon
0
0
0
0
0
0
0
0
0
0
cumultive f graph measures
0
0
0
0
0
0
0
0
0
0
uppper boundarycumulative frequency
delivery timefrequencyless than 550
5 -< 103less than 10103
10 -
Unequal class sizes
Delivery timeNumber of deliveriesFrequency densityAdjusted height using standard class width 50-
Blank grids
Chart2
0
0
0
0
0
0
0
0
0
0
histogram equal
50
53Histogram equal classes
103
107
157
1510
2010
2012
2512
258
308
306
356
352
402
402
452
450
histogram equal
Delivery times (days)
Frequency
Histogram of delivery times
histogram1
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0
time (days)
frequency
histogram of delivery times
blank grids
50
100
150
200
250
300
350
400
450
500
blank grids
histograms2
cumultive frequencytable
cumultive frequencytable
0
0
0
0
0
0
0
0
0
0
cumultive f graph
0
0
0
0
0
0
0
0
0
0
cumultive f graph measures
uppper boundarycumulative frequency
delivery timefrequencyless than 550
5 -< 103less than 10103
10 -
Histogram standard class size 5
Chart4
1.5
1.5
1.5
7
7
10
10
12
12
8
8
3.3
3.3
3.3
3.3
0
delivery times (days)
frequency per class size 5
histogram using standard class size 5
histogram equal
50
53Histogram equal classes
103
107
157
1510
2010
2012
2512
258
308
306
356
352
402
402
452
450
histogram equal
Delivery times (days)
Frequency
Histogram of delivery times
histogram1
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0
time (days)
frequency
histogram of delivery times
blank grids
50
100
150
200
250
300
350
400
450
500
01.5
51.5
101.5
107
157
1510
2010
2012
2512
258
308
303.3
353.3
403.3
453.3
450
blank grids
histograms2
cumultive frequencytable
delivery times (days)
frequency per class size 5
histogram using standard class size 5
cumultive f graph
cumultive f graph
0
0
0
0
0
0
0
0
0
0
cumultive f graph measures
0
0
0
0
0
0
0
0
0
0
uppper boundarycumulative frequency
delivery timefrequencyless than 550
5 -< 103less than 10103
10 -
Histogram using Frequency density
Chart6
0.3
0.3
0.3
1.4
1.4
2
2
2.4
2.4
1.6
1.6
0.7
0.7
0.7
0.7
0
delivery times (days)
frequency density
histogram using frequency density
histogram equal
50
53Histogram equal classes
103
107
157
1510
2010
2012
2512
258
308
306
356
352
402
402
452
450
histogram equal
Delivery times (days)
Frequency
Histogram of delivery times
histogram1
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0
time (days)
frequency
histogram of delivery times
blank grids
50
100
150
200
250
300
350
400
450
500
01.5
51.5
101.5
107
157
1510
2010
2012
2512
258
308
303.3
353.3
403.3
453.3
450
00.3
50.3
100.3
101.4
151.4
152
202
202.4
252.4
251.6
301.6
300.7
350.7
400.7
450.7
450
blank grids
histograms2
cumultive frequencytable
delivery times (days)
frequency per class size 5
histogram using standard class size 5
cumultive f graph
delivery times (days)
frequency density
histogram using frequency density
cumultive f graph measures
cumultive f graph measures
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
uppper boundarycumulative frequency
delivery timefrequencyless than 550
5 -< 103less than 10103
10 -
Guidelines for grouping dataSo, in the example,Largest observation = 41 Smallest = 5Require say 8 classesclass width =
Cumulative frequency table
Delivery timeFrequency CumulativeFrequencyLess than 505-
Cumulative frequency graph
Finding measures from the cumulative frequency graph
Measures for this examplemedian look at cumulative frequency of 25 on graph22 daysupper quartile -cumulative frequency of 37.5 on graph28 dayslower quartile - cumulative frequency of 12.5 on graph16 daysinter-quartile range is UQ LQ 28-16=12 days20th percentile look at cumulative frequency of 10 on graph 15 dayslook at 30 days horizontal axis to give 40 deliveries so 50-40 =10 deliveries are more than 30 days90% of deliveries take less 34 days
Measures of locationThe mode : Most frequently occurring item 35
The median: Middle number. The mean
28 28 35 35 35 36 39 44 44 50
Using frequency formula
XFFX2825635310536136391394428850150
Measure from a grouped frequency table
TimeFrequencyFMidpointXFXCumulative frequency0-
MeasuresMean =
Mode is estimated to be 22.5, the middle of the modal class
Median
Which measure is best3 33 45 710 10 10 25 40Mean= 10.9~11Mode= 3, 10Median = 7
quartilesLower quartileUpper quartile
Measures of spread
ArmstrongBarrett3 6 3 4 4 6 4 2 4 5 3 5 4 4 3 5Ordered2 3 3 4 4 4 6 6 3 3 4 4 4 4 5 5Mean4 weeks4Mode44Median44ConcludeLittle differenceRange6-2 = 45-3=2Inter-quartile rangeStandard deviationCoefficient of variation
Standard deviation
X23344466-2-1-1000224110004414
1.32
Coefficient of variationThe higher the ratio, the greater the spread around the mean. Lengths mean=55standard deviation = 28.7coefficient of variation=52%Weightsmean=5.5standard deviation = 2.8.7coefficient of variation = 52%
Mean and Standard Deviation for Armstrong
XFFXFX20.51473.51.51522.533.752.51845112.53.516561964.51567.5303.755.51160.5332.756.51171.5464.75totals3301447
mean
Standard deviation
Probabilty examplesExamplesThrow a coin. The probability of a head = 0.5
There are three counters in a bag, one red, one green and one blue. One counter is pulled out.The probability that the counter is red =
The counter is then replaced and a second pulled out.List all the outcomes: RR, RG, RB, GR, GG, GB, BR, BG, BB
the probability that both the first and the second were red =
exampleOver the last month (November) a machine has broken down on three days. What is the probability the machine breaks down?
exampleIn a sample of adults these probabilities were found: P(male) = 0.5P(Married)=0.6P(full time job)=0.9A person is selected at random. What is the probability that the person ismarried and male
ii) male and in a full time job
iii) female 0.5
exampleP(male)=0.7P(aged 40 to 59)=0.4P(aged 60 to 69)=0.15P(aged 70 or more)=0.1
Female1-0.7 = 0.3
Probability(40 to 59) or (60 to 69) years old0.4+0.15=0.55
Probability (female) and (40 to 59) years old0.30.4 = 0.12
male or aged 40 to 59? CANNOT SAY NOT 0.7 + 0.4 = 1.1
exampleThe probability that firm A makes a profit has been assessed to be 0.6. The probability that the firm breaks even is 0.3.What is the probability that the firm makes a loss?1- 0.6 - 0.3 = 0.1
example
Firm ProfitBreak evenlossA0.60.30.1B0.70.10.2
exampleboth firms make a profit 0.60.7=0.42firm A does not make a profit 1-0.6=0.4firm A makes a profit or breaks even0.6+0.3 = 0.9firm B does not make a profit 0.3neither firms make a profit 0.40.3=0.12at least one firm makes a profit1-0.12=0.88only one firm makes a profit 1- 0.42 0.12 = 0.46
Throw a dieExpected score is
expectationExpected number of minutes late00.4 + 30.3 + 50.15 + 100.1 + 150.05 = 3.4 minutes
03510150.40.300.150.10.05
Spread of valueswith =3000 hrs and =200 hrsapproximately 68% of the bulbs will last between 2800 hours and 3200 hours,
approximately 95% of the bulbs will last between 2600 hours and 3400 hours,
approximately 99.75% of the bulbs will last between 2400 hours and 3600 hours.
Using normal tablesiP(Z1.3) Read directly from the table =0.0968
iiiP(Z
Find Kfind k such that P(Z>K) = 0.15
also means P(Z
Solution to example = 2000 hours and = 250 hours(a) less than 1750 hours
from tables area to the left of -1 = 0.1587
more than 2350 hours,
from tables area to the left of -1.4 = 0.0808
Example (c)between 1800 hours and 2400 hours?
area to the left of -0.8= 0.2119
area to left of 1.6 = 1- 0.0548 = 0.9452
area in between = 0.9452-0.2119 = 0.7333
Example(d)4% fail
P(Z
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