agenda informationer –g-mails til klaus –brug supplerende litt. –Åben excel (opsamling fra...
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Agenda
• Informationer– G-mails til Klaus– Brug supplerende litt.– Åben Excel
• (Opsamling fra sidst)• Deskriptiv statistik
– Teori– Små øvelser i Excel
• Videre med projekt 1– BPN datasæt
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Opsamling fra sidst: Opgave 1
• Identify the variable type as either categorical or quantitative
1. Number of siblings in a family
2. County of residence
3. Distance (in miles) to school
4. Marital status
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Opsamling fra sidst: Opgave 2
• Identify each of the following variables as continuous or discrete
1. Length of time to take a test
2. Number of people waiting in line
3. Number of speeding tickets received last year
4. Your dog’s weight
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Proportion & Percentage (Relative Frequencies)
• The proportion (andelen) of the observations that fall in a certain category is the frequency (count) of observations in that category divided by the total number of observations
• Frequency of that class Sum of all frequencies
• The Percentage is the proportion multiplied by 100. Proportions and percentages are also called relative frequencies.
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Frequency, Proportion, & Percentage Example
• If 4 students received a “10” out of 40 students, then,– 4 is the frequency
– 4/40 = 0,10 is the proportion (and relative frequency)
– 10% is the percentage 0,1*100=10%
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Frequency Table
• A frequency table is a listing of possible values for a variable, together with the number of observations and / or relative frequencies for each value
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Opgave #3• Et webbureau holder øje med antallet af billeder på en
række toneangivende hjemmesider. I en periode har man først statistik med, om antallet af billeder på forsiden er højere, uændret eller mindre. Resultaterne var:
– Flere: 21– Uændret: 7– Færre: 12
• Spørgsmål:1. Navngiv den variabel webbureauet interesserer sig for?2. Hvilken slags variabel er det?3. Udregn andelene i Excel
• Data til næste opgave i Excel: http://www.dst.dk/da/Statistik/emner/navne/Baro.aspx
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Exploring Data with Graphs and Numerical Summaries
1. Calculating the mean
2. Calculating the median
3. Comparing the Mean & Median
4. Definition of Resistant
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Mean (gennemsnittet)
• The mean is the sum of the observations divided by the number of observations– n betegner antallet af observationer (=stikprøvestørrelsen)
– y1, y2, y3, … yi ,..., yn betegner de n observationer
– betegner gennemsnittet
• It is the center of mass. Do it in Excel
n
y
n
yyyy i in
21
y
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Median
• The median is the midpoint of the observations when they are ordered from the smallest to the largest (or from the largest to smallest)
• Order observations• If the number of observations is:
– Odd, then the median is the middle observation– Even, then the median is the average of the two
middle observations
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Median
n = 10 (n+1)/2 = 5,5
Median = (99+101) /2 = 100
2,b) If n is even, the median is the mean of the two middle observations
n = 9 (n+1)/2 = 10/2 = 5 Median = 99
2,a) If n is odd, the median is observation (n+1)/2 down the list
1) Sort observations by size,n = number of observations
______________________________Order Data1 78 2 91 3 94 4 98 5 99 6 101 7 103 8 105 9 114
Order Data1 78 2 91 3 94 4 98 5 99 6 101 7 103 8 105 9 114
10 121
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Find the mean and median in ExcelDet gns. tidsforbrug (sek.), som det tager at indlæse nyheder på en app fra 8 nyhedssider:
2,3 1,1 19,7 9,8 1,8 1,2 0,7 0,2
Hvad er mean og median?
a. Mean = 4,6 Median = 1,5
b. Mean = 4,6 Median = 5,8
c. Mean = 1,5 Median = 4,6
Hvad sker der med gns. og median, hvis den største observation ganges med 10?
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Comparing the Mean and Median
• The mean and median of a symmetric distribution are close together,
• For symmetric distributions, the mean is typically preferred because it takes the values of all observations into account
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Comparing the Mean and Median• In a skewed distribution, the mean is
farther out in the long tail than is the median– For skewed distributions the median is
preferred because it is better representative of a typical observation
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Resistant Measures
• A numerical summary measure is resistant if extreme observations (outliers) have little, if any, influence on its value– The Median is resistant to outliers– The Mean is not resistant to outliers
• Hvis I kun kan få én oplysning om løn-niveauet i en virksomhed med 4 ansatte, vil I så have gennemsnit eller median?
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Exploring Data with Graphs and Numerical Summaries
1. Calculate the range (variationsbredden)
2. Calculate the standard deviation
3. Know the properties of the standard deviation, s
4. Know how to interpret the standard deviation, s: The Empirical Rule
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Range
• One way to measure the spread is to calculate the range. The range is the difference between the largest and smallest values in the data set;
Range = max min
• The range is strongly
affected by outliers
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Standard DeviationObservations Deviations Squared deviations
1792 17921600 = 192 (192)2 = 36,864
1666 1666 1600 = 66 (66)2 = 4,356
1362 1362 1600 = -238 (-238)2 = 56,644
1614 1614 1600 = 14 (14)2 = 196
1460 1460 1600 = -140 (-140)2 = 19,600
1867 1867 1600 = 267 (267)2 = 71,289
1439 1439 1600 = -161 (-161)2 = 25,921
Sum = 0 Sum = 214,870
xxi ix 2xxi
67.811,3517
870,2142
s24.18967.811,35 s
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Standard Deviation
• Find the mean• Find the deviation of each value from the mean• Square the deviations• Sum the squared deviations• Divide the sum by n-1
1
)( 2
n
xxs
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Standard Deviation
• Gives a measure of variation by summarizing the deviations of each observation from the mean and calculating an adjusted average of these deviations.
1
)( 2
n
xxs
Site
Obs.1 2
3 Sum n Gns. Std.afv.
U 5 5 5 15 3 5 0,0
V 4 5 6 15 3 5 1,0
X 3 5 7 15 3 5 2,0
Y 2 5 8 15 3 5 3,0
Z 1 5 9 15 3 5 4,0
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Properties of the Standard Deviation
• s measures the spread of the data
• s = 0 only when all observations have the same value, otherwise s > 0, As the spread of the data increases, s gets larger,
• s has the same units of measurement as the original observations. The variance=s2 has units that are squared.
• s is not resistant. Strong skewness or a few outliers can greatly increase s.
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Magnitude of s: Empirical Rule
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Exploring Data with Graphs and Numerical Summaries
1. Obtaining quartiles and the 5 number summary
2. Calculating interquartile range and detecting potential outliers
3. Drawing boxplots
4. Comparing Distributions
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Percentile
• The pth percentile is a value such that p percent of the observations fall below or at that value
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Finding Quartiles
• Splits the data into four parts– Arrange the data in order
– The median is the second quartile, Q2
– The first quartile, Q1, is the median of the lower half of the observations
– The third quartile, Q3, is the median of the upper half of the observations
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M = median = 3,4
Q1= first quartile = 2,2
Q3= third quartile = 4,35
1 0.62 1.23 1.64 1.95 1.56 2.17 2.38 2.39 2.5
10 2.811 2.912 3.313 3.414 3.615 3.716 3.817 3.918 4.119 4.220 4.521 4.722 4.923 5.324 5.625 6.1
Measure of spread: quartiles
The first quartile, Q1, is the value in the sample
that has 25% of the data at or below it and 75%
above
The second quartile is the same as the median of
a data set, 50% of the obs are above the median
and 50% are below
The third quartile, Q3, is the value in the sample
that has 75% of the data at or below it and 25%
above
Quartiles divide a ranked data set into four equal parts.
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Quartile Example
Find the first and third quartiles
Number of downloaded apps the last 10 days:
2 4 11 13 14 15 31 32 34 47
What is the correct answer?
a. Q1 = 2 Q3 = 47
b. Q1 = 12 Q3 = 31
c. Q1 = 11 Q3 = 32
d. Q1 =12 Q3 = 33
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Calculating Interquartile range
• The interquartile range is the distance between the third quartile and first quartile:
• IQR = Q3 Q1
• IQR gives spread of middle 50% of the data
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Criteria for identifying an outlier
• An observation is a potential outlier if it falls more than 1,5 x IQR below the first quartile or more than 1,5 x IQR above the third quartile
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5 Number Summary
• The five-number summary of a dataset consists of the– Minimum value– First Quartile– Median– Third Quartile– Maximum value
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Boxplot• A box goes from the Q1 to Q3• A line is drawn inside the box at the median• A line goes from the lower end of the box to the smallest
observation that is not a potential outlier and from the upper end of the box to the largest observation that is not a potential outlier
• The potential outliers are shown separately
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Comparing DistributionsBox Plots do not display the shape of the distribution as clearly as
histograms, but are useful for making graphical comparisons of two
or more distributions
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Exploring Data with Graphs and Numerical Summaries
1. Distribution
2. Graphs for categorical data: Bar graphs and pie charts
3. Graphs for quantitative data: Dot plot, stem-leaf, and histogram
4. Constructing a histogram
5. Interpreting a histogram
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Distribution (Fordeling)
• A graph or frequency table describes a distribution
• A distribution tells us the possible values a variable takes as well as the occurrence of those values (frequency or relative frequency)
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Graphs for Categorical Variables
• Use pie charts and bar graphs to summarize categorical variables– Pie Chart: A circle having a “slice of pie” for
each category– Bar Graph: A graph that displays a vertical
bar for each category
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Pie Charts
• Pie charts:– Used for summarizing a categorical variable– Drawn as a circle where each category is
represented as a “slice of the pie” – The size of each pie slice is proportional to
the percentage of observations falling in that category
– Draw a pie chart for the web bureau data!
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Bar Graphs
• Bar graphs are used for summarizing a categorical variable
• Bar Graphs display a vertical bar for each category
• The height of each bar represents either counts (“frequencies”) or percentages (“relative frequencies”) for that category
• Usually easier to compare categories with a bar graph than with a pie chart
• Draw a bar graph for the web bureau data!
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Histograms• A histogram is a graph that uses bars to portray
the frequencies or the relative frequencies of the possible outcomes for a quantitative variable.
• Make a histogram with 12 observations
1%1%
4%
8%
10%
12%
15%
13%13%
9%
5% 5%
2%1%
1% 0%1% 1%
3%
7%
3%
11%
19% 19%
9%8%
4%
11%
2%1% 1%
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Population Stikprøve
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Interpreting Histograms
• Overall pattern consists of center, spread, and shape– Assess where a distribution is centered by
finding the median (50% of data below median 50% of data above),
– Assess the spread of a distribution, – Shape of a distribution: roughly symmetric,
skewed to the right, or skewed to the left
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A distribution is skewed to the left if the left tail is longer than the right tail
Shape
• Symmetric Distributions: if both left and right sides of the histogram are mirror images of each other
A distribution is skewed to the right if the right tail is longer than the left tail
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Examples of Skewness
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Shape and Skewness
• Consider a data set of the scores of students on a very easy exam in which most score very well but a few score very poorly:
• What shape would you expect a histogram of this data set to have?
a. Symmetric
b. Skewed to the left
c. Skewed to the right
d. Bimodal