Get documents about "Logs - ROHAN Academic Computing - SDSU
Document Sample
Logarithms
This series of slides demonstrates the usage of logarithms. Logs have a
variety of applications in economics and finance.
To obtain the spreadsheet that contains the examples click on the following:
ftp://rohan.sdsu.edu/faculty/vandenberg/Logs.xls
ms. Logs have a
lick on the following:
How do logarithms work?
ROW/COL B C D E
4
5
6 Number 1 Number 2 Result of Multiplying
7 Number 10 10 100
8 Power of the number 1 1 2
9
10 Notice we get the same result by multiplying the numbers as we do by adding the power of the numbers.
11 The power of the number is also its exponent. By default we do not write the power if it is 1.
12
13 Works for any numbers Result of Multiplying
14 Number 100 10 1000
15 Power of the number 2 1 3
16
17 We also that we can write: 1,000,000
18 as #NAME?
19 or =10^6 (remember an ^ means raised to the power)
20
21 Thus
22 Number 1,000,000 1,000,000 1,000,000,000,000
23 Power 6 6 12
24
25 In fact the spread sheet might use: 1E+12 to represent the number.
26 As you can see the E+12 means move the decimal point 12 space to the right and you get the answer.
27
28 It would not be hard to figure the exponent for any number which is an even power of 10
29
30 So then we can see that 100,000
31 is the same as #NAME?
32 or =10^5
33
34 Rather than count the number of 10's need to get the number a spreadsheet provides a function to do this
35
36
Log of number
(or the number
37 Number of 10s) Formula
38 10 1 #NAME?
39 100 2 #NAME?
40 1000 3 #NAME?
41 10000 4 #NAME?
42 100000 5 #NAME?
43 1000000 6 #NAME?
44
45 The log function is particularly helpful if we wish to find the power of 10 when the number is not exactly a
46 We can write 12,345
47 as =10^4.09149109426795
48 or the log of 12,345 = 4.09149109426795
49
50 Works for any numbers
51 Number 1 Number 2 Result of Multiplying
52 Number 12,345 678 8,369,910
53 Power of the number 4.091491094 2.831229694 6.922720788
54
55
56 In days before electronic calculators and computers logs were commonly used because addition is easier
57 Tables of logarithms were available as were slide rules which were base on logs.
58
59 To reverse the process, to convert a logarithm into a number, you need to do the opposite.
60 Again for even powers of 10 that is not difficult. For uneven powers of 10 use the spreadsheet power symb
61
62 Log of a number is: 4.091491094
63 Exponent form: =10 ^ 4.09149109426795
64 The number is 12,345 #NAME?
65
66 Therefore 10^(log(Number)) = Number
67 Example Number
68 12,345 12,345 #NAME?
F G H I
Mathematically we can say:
=10*10
=1+1
adding the power of the numbers.
te the power if it is 1.
Mathematically we can say:
=100*10
=2+1
an ^ means raised to the power)
=1000000*1000000
=6+6
nt the number.
e right and you get the answer.
even power of 10
sheet provides a function to do this
0 when the number is not exactly a power of 10
Mathematically we can say:
=12345*678
=4.09149109426795+2.83122969386706
nly used because addition is easier than multiplication
to do the opposite.
10 use the spreadsheet power symbol (^) to do it.
Log Scales
ROW/COL B C D E F G
4 Frequently data is hard to understand without using logs
5 Here is some Forecasted data for the next few years. Look at the graphs below
6
7
Alternative
log of way to
8 Year Forecast number express the # #NAME?
9 2000 10 1 10 ^1
10 2001 100 2 10 ^2
11 2002 1,000 3 10 ^3
12 2003 10,000 4 10 ^4
13 2004 100,000 5 10 ^5
14 2005 1,000,000 6 10 ^6
15 2006 10,000,000 7 10 ^7
16 2007 100,000,000 8 10 ^8
17 2008 1,000,000,000 9 10 ^9
18 2009 10,000,000,000 10 10 ^10
19 2010 100,000,000,000 11 10 ^11
20 2011 1,000,000,000,000 12 10 ^12
21
22
23
24 Here is one picture of the data
25 Notice that it is hard to see what is going on.
26
27
28 Original Data
29
30
31 1,200,000
Millions
32
33 1,000,000
34
35
36 800,000
37
38 600,000
Values
39
40
41
400,000
42
43 200,000
44
45 0
46
47 1995 2000 2005 2010 2015
48 -200,000
49
50
51
Time
52
53
54
55 Here is a picture using the log scale
56 Notice that you can see the series is going straight up.
57
58
59
60
61 Using the log scale of the Y axis
62
63
64
65
66 1,000,000,000,000 10 ^12
67 100,000,000,000 10 ^11
68
10,000,000,000 10 ^10
log values are graphed
69
70 1,000,000,000 10 ^9
71 100,000,000 10 ^8
72 10,000,000 10 ^7
73
74
1,000,000 10 ^6
75 100,000 10 ^5
76 10,000 10 ^4
77 1,000 10 ^3
78
79
100 10 ^2
80 10 10 ^1
81 1
82
83
1995 2000 2005 2010 2015
84 Time
85
86
87
88 Using the logs of the data rather than a log axis
89 You see the same effects
90
91
92
93
Transformed data
94
95
96 14
97
98 12 10 ^12
the log of the values
99 10 ^11
100
10 10 ^10
101
10 ^9
Using the log of the values
102 10 ^9
103 8 10 ^8
104 10 ^7
105 6 10 ^6
106
107
10 ^5
108 4 10 ^4
109 10 ^3
110 2 10 ^2
111 10 ^1
112
113
0
114 1998 2000 2002 2004 2006 2008 2010 2012
115 Time
H I
2015
This result was
achieved by right
clicking the axis and
then choosing to
check this log box.
10 ^12
2012
The famous log scale.
If you live in California you should be familiar with a famous log scale.
It is called the Richter Scale, it measures an earthquake's power.
So an earthquake with a Richter reading of 4
has a certain power. If another earthquake with a Richter reading of 5
occurs, how much more powerful is it.
Since Richter readings are in log form then a 5 is 10 times more powerful than a 4
So the difference between a 3 and a 7 magnitude is substantial.
A 7 is 10,000 more powerful than a 3.
For more information try:
http://neic.usgs.gov/neis/general/handouts/richter.html
owerful than a 4
Other bases for logs.
ROW/COL B C D E F G H I
4
5
6 There is nothing magic about base 10 (or common logarithms). You could build logs on any base you want.
7 In fact the log function has an option to set the base to any you may want to use.
8
9 Number log to the base 2 of the number
10 2 1 =2 ^ 1
11 4 2 =2 ^ 2
12 8 3 =2 ^ 3
13 16 4 =2 ^ 4
14 32 5 =2 ^ 5
15 64 6 =2 ^ 6
=log(number, base)
16
17 Other bases
18 Base
19 Number 2 5 10 16
20 2 1.0000 0.4307 0.3010 0.2500 #NAME?
21 4 2.0000 0.8614 0.6021 0.5000
22 5 2.3219 1.0000 0.6990 0.5805
23 10 3.3219 1.4307 1.0000 0.8305
24 16 4.0000 1.7227 1.2041 1.0000
25 20 4.3219 1.8614 1.3010 1.0805
26 #NAME?
27 The blue numbers are the logs of the number for different bases
28 Notice that when the base is equal to the number the log is always 1
29
30 Natural logarithms
31
32 Look at the following expansion:
33 X (1+1/X)^X
34 1 2.00000 As X expands the
35 2 2.25000 value of expression
36 3 2.37037 increases, but at a
37 4 2.44141 slower and slower
38 5 2.48832 rate and reaches a
39 6 2.52163 limit. This limit is a
40 7 2.54650
number which like
41 8 2.56578
pi, never rounds off
42 9 2.58117
even. This number
43 10 2.59374
is called "e." It has a
44 1010 2.71694
lot of useful
45 2010 2.71761
46 3010 2.71783 mathematical
47 4010 2.71794 properties. Because
48 5010 2.71801 of this it is used
49 6010 2.71806 frequently. As an
50 7010 2.71809 example the
51 8010 2.71811 derivative of e^x is
e^x. Very
convenient.
mathematical
properties. Because
of this it is used
frequently. As an
example the
derivative of e^x is
52 9010 2.71813 e^x. Very
53 10010 2.71815 convenient.
54 11010 2.71816
55 12010 2.71817
56 very large 2.7182818284591 <==Value of "e" used by Excel
57 1000 #NAME?
58 Natural logarithms use "e" as their base.
59 So the log to the base e is the value you need to raise e to get the number
60
61 Spreadsheets use a special set of functions to manipulate the natural logs.
62
63 To get the natural logarithm use =ln(number)
64 To get the anti-logarithm (to reverse the process) use =exp(mumber)
65
66
67 Number ln(number) Result
68 1 0.000000 = 2.71828182845905 ^ 0 = 1
69 2 0.693147 = 2.71828182845905 ^ 0.693147180559945 = 2
70 5 1.609438 = 2.71828182845905 ^ 1.6094379124341 = 5
71 7 1.945910 = 2.71828182845905 ^ 1.94591014905531 = 7
72 10 2.302585 = 2.71828182845905 ^ 2.30258509299405 = 10
73 25 3.218876 = 2.71828182845905 ^ 3.2188758248682 = 25
74
75 So again exp(ln(number)=number
76
77 Number
78 1000 1000 #NAME?
79
80
81
82
83
gs on any base you want.
Natural logs are frequently used in finance.
ROW/COL B C D E F G H
5
6
7 The compound value of a $ is:
8
9
10
11
12 V = P *(1+ i/n)^(n*t)
13
14 where:
15
16 V is the ending value
17 P is the present value you start with
18 I is the rate of interest
19 n is the frequency of compounding
20 t is the length of time
21
22 Notice that part of the formula looks a lot like the expansion to get the value of "e"
23 If we let P=1 and t=1 it is exactly the same formula.
24
25
26
27
28 V = 1 *(1+ i/n)^(n*1)
29
30 If n =1 then we get: (which is frequently the form of the compound amount formula)
31
32
33
34
35 V = P *(1+ i/1)^(1*t)
36 Impact of compounding
37 Using an interest rate of: 5.0000%
38 Assume the time period is 1 year and the amount is $1
39
40 Frequency of compounding n V Annual Rate of return
41 Annual 1 $1.05000 5.0000%
42 Semi-Annual 2 $1.05063 5.0625%
43 Quarterly 4 $1.05095 5.0945%
44 Monthly 12 $1.05116 5.1162%
45 Daily 365 $1.05127 5.1267%
46 Hourly 8760 $1.05127 5.1271%
47 Minutely 525600 $1.05127 5.1271%
48 Secondly 31536000 $1.05127 5.1271%
49 Infinite =EXP(i) $1.05127 5.1271%
50
51 Notice that as the frequency of compounding increase our rate of return increase, but at a decreasing rate.
52 Until the difference between daily, secondly and infinite is nil.
53
54
55 So if you allow infinite compounding the future value of a dollar formula becomes:
56
57
58
59
60 V = P*exp(i*t)
61
62 Remembering that ln and e^x are opposite transactions we can determine an equivalence between rates of i
63
64
65 You would indifferent between the following rates compounded once a year, or the rates compounded conti
Compounded Compounded Compound amount of $1
66 once a year continuously at the end of a year
ln(1+Annual
Annual Rate of Rate of Once a
67 return Return) year Continuously
68 5.0000% 4.8790% $1.0500 $1.0500
69 5.0625% 4.9385% $1.0506 $1.0506
70 5.0945% 4.9690% $1.0509 $1.0509
71 5.1162% 4.9896% $1.0512 $1.0512
72 5.1267% 4.9997% $1.0513 $1.0513
73 5.1271% 5.0000% $1.0513 $1.0513
74
75 You can always find a rate that will cause you to be indifferent between an annual rate and continuously com
76
Continuously
77 Annual Rate Compounded
78 5.0000% 4.8790% #NAME?
79
80 Or you find a continuously compounded rate that is equal to an annual rate by using exp(i)-1
81
Continuously
82 Compounded Annual Rate
83 5.5000% 5.6541% #NAME?
84
85 Try it by changing the green numbers
86
87
88 In many cases theoretical consideration require that we use continuously compounded rates in finance.
89 In general when using the CAPM and Black/Scholes you should use continuously compounded rates.
90
I
ln(Annual Rate of Return)
4.8790%
4.9385%
4.9690%
4.9896%
4.9997%
5.0000%
5.0000%
5.0000%
5.0000%
e, but at a decreasing rate.
uivalence between rates of interest and compounding
the rates compounded continuously.
al rate and continuously compounded one by taking the ln(1+i)
sing exp(i)-1
ounded rates in finance.
ly compounded rates.
The fundamental equation of finance:
ROW/COL B C D E F G H I
5
6
7 The compound amount equation is:
8
9
10
11 V = P *(1+ i)^t
12
13 This equation can also be stated as follows.
14
15
16
17 ln(V )= ln(P)+t*ln(1+ i)
18
19
20 I= 10.00%
21 P= $100.00
22 t= 10
23
24 ln(v)= 5.558272 #NAME?
25
26 V= $259.37 #NAME?
27
28 Using the FV function $259.37 #NAME?
29
30 An alternative would be:
31
32
33
34
35
36 ln($1.00 )=t*ln(1+ i)
37
38
39 I= 10.00%
40 P= $1.00
41 t= 10
42
43 ln(1)= 0.9531018 #NAME?
44 FV Factor 2.5937425 #NAME?
45
46 For $100
47 V= $259.37 #NAME?
48
49
50 Another alternative for annual compounding would be:
51
52
53
54
55
56 V=P*exp(t*ln(1+ i))
57
58 I= 10.00%
59 P= $100.00
60 t= 10
61
62 V= $259.37 #NAME?
63
64 For continuous compounding the formula would be:
65
66
67
68
69 V=P*exp(t*i)
70
71 I (continuously) = 10.00%
72 P= $100.00
73 t= 10
74
75 V= $271.83 #NAME?
76
77
78 Note the difference of $12.45 #NAME?
79 is caused by the frequency of compounding
80
81
82
83
84
Stock Returns
ROW/COL B C D E F G
3
4 As you know many things in this world are normally distributed. This is very convenient for statistical analy
5
6 Unfortunately stock returns are not normally distributed. You can described the reason in a variety of ways
7
8 However, there is a fair amount of evidence that stock returns are approximately normally distributed if you
9 To do this is fairly simple.
10
11
12 Below is the data for IBM from Feb, 1998 to May, 2003
The blue numbers are the ln returns which are in general the ones you want to use for analysis such as com
13 Beta's and it is also the data needed to implement the Option Pricing models.
Adjusted Cash 1+ Return =
14 Date Close Dividend (Pt+Dt)/Pt-1 ln(1+Return)
15 #NAME?
16 Feb-98 51.81
17 Mar-98 51.53 0.99460 -0.542% #NAME?
18 Apr-98 57.48 1.11547 10.927%
19 May-98 58.29 1.01409 1.399%
20 Jun-98 56.96 0.97718 -2.308%
21 Jul-98 65.73 1.15397 14.321%
22 Aug-98 55.87 0.84999 -16.253%
23 Sep-98 63.75 1.14104 13.194%
24 Oct-98 73.67 1.15561 14.463%
25 Nov-98 81.92 1.11199 10.615%
26 Dec-98 91.47 1.11658 11.027%
27 Jan-99 90.91 0.99388 -0.614%
28 Feb-99 84.21 0.92630 -7.656%
29 Mar-99 87.93 1.04418 4.323%
30 Apr-99 103.78 1.18026 16.573%
31 May-99 115.09 1.10898 10.344%
32 Jun-99 128.24 1.11426 10.819%
33 Jul-99 124.71 0.97247 -2.791%
34 Aug-99 123.59 0.99102 -0.902%
35 Sep-99 120.06 0.97144 -2.898%
36 Oct-99 97.48 0.81193 -20.834%
37 Nov-99 102.26 1.04904 4.787%
38 Dec-99 107.03 1.04665 4.559%
39 Jan-00 111.37 1.04055 3.975%
40 Feb-00 101.95 0.91542 -8.838%
41 Mar-00 117.45 1.15204 14.153%
42 Apr-00 110.63 0.94193 -5.982%
43 May-00 106.47 0.96240 -3.833%
44 Jun-00 108.71 1.02104 2.082%
45 Jul-00 111.37 1.02447 2.417%
46 Aug-00 130.99 1.17617 16.226%
47 Sep-00 111.74 0.85304 -15.895%
48 Oct-00 97.73 0.87462 -13.397%
49 Nov-00 92.77 0.94925 -5.209%
50 Dec-00 84.34 0.90913 -9.527%
51 Jan-01 111.13 1.31764 27.584%
52 Feb-01 99.12 0.89193 -11.437%
53 Mar-01 95.43 0.96277 -3.794%
54 Apr-01 114.24 1.19711 17.991%
55 May-01 110.93 0.97103 -2.940%
56 Jun-01 112.61 1.01514 1.503%
57 Jul-01 104.39 0.92700 -7.580%
58 Aug-01 99.17 0.95000 -5.130%
59 Sep-01 91 0.91762 -8.598%
60 Oct-01 107.23 1.17835 16.412%
61 Nov-01 114.69 1.06957 6.726%
62 Dec-01 120.02 1.04647 4.543%
63 Jan-02 107.05 0.89193 -11.436%
64 Feb-02 97.36 0.90948 -9.488%
65 Mar-02 103.19 1.05988 5.816%
66 Apr-02 83.11 0.80541 -21.641%
67 May-02 79.82 0.96041 -4.039%
68 Jun-02 71.44 0.89501 -11.092%
69 Jul-02 69.85 0.97774 -2.251%
70 Aug-02 74.96 0.15 1.07530 7.260%
71 Sep-02 57.98 0.77348 -25.686%
72 Oct-02 78.5 1.35392 30.300%
73 Nov-02 86.59 0.15 1.10497 9.982%
74 Dec-02 77.21 0.89167 -11.466%
75 Jan-03 77.91 1.00907 0.903%
76 Feb-03 77.81 0.15 1.00064 0.064%
77 Mar-03 78.29 1.00617 0.615%
78 Apr-03 84.74 1.08239 7.917%
79 May-03 86.18 1.01699 1.685%
80
81 Monthly Average Return = 0.817% #NAME?
82 Monthly Standard Deviation = 11.317% #NAME?
83
84 Annual Average Return = 9.800% #NAME?
85 Annual Standard Deviation = 39.203% #NAME?
86
87
H I
y convenient for statistical analysis.
d the reason in a variety of ways but the basic reason is that stock prices cannot be negative.
mately normally distributed if you take the log (natural) of the return.
t to use for analysis such as computing
ls.
Other applications of logarithms: Trends
ROW/COL B C D E F G H I
5 Other applications of logarithms
6
7 Suppose you have the following series of numbers for which you are trying to forecast the next one
8 10%
9 Year Sales Year ln(sales)
10 1 $100.00 1 4.6052 #NAME?
11 2 $110.00 2 4.7005 #NAME?
12 3 $121.00 3 4.7958 #NAME?
13 4 $133.10 4 4.8911 #NAME?
14 5 $146.41 5 4.9864 #NAME?
15 6 $161.05 6 5.0817 #NAME?
16 7 $177.16 7 5.1770 #NAME?
17 8 $194.87 8 5.2723 #NAME?
18 9 $214.36 9 5.3677 #NAME?
19 10 $235.79 10 5.4630 #NAME?
20 11 $259.37 11 5.5583 #NAME?
21 12 $285.31 12 5.6536 #NAME?
22 13 ? 13 ?
23 The linear trend forecasts:
24 Forecast Series ln(series) Exp(ln forecast)
25 13 $286.25 13 5.7489 $313.84 #NAME?
26 #NAME? #NAME?
27
28 It is clear that the process is not linear. Thus a linear estimate would not be expected to produce good estim
29
30
31
32 Sales
33
34
$350.00 6.0000
35
36 Forecast 5.8000
$300.00
37 5.6000
38
$250.00 5.4000
39
Sales ($)
40 5.2000
ln(Sales)
$200.00
41
5.0000
42 $150.00 Trend Line
43 4.8000
44 $100.00 4.6000
45
Raw Data 4.4000
46 $50.00
47 4.2000
48 $0.00 4.0000
49 0 5 10 15
0 5 10 15
50
Time
51
52
53
54
ast the next one
d to produce good estimate
Natural Log of Sales
6.0000
Forecast
5.8000
5.6000
5.4000
5.2000
Trend Line
5.0000
4.8000
4.6000
4.4000
4.2000
4.0000
0 5 10 15
0 5 10 15
Time
Multiplicative Demand Functions
ROW/COL B C D E F G
5 For demand equations we can also use the following:
6
7 Q = aPcAdYe
8 This is called a multiplicative demand function.
9 The exponents are the elasticities of the variables.
10
11
12 Ln(Q) = a + c*ln(P) + d*ln(A) + e*ln(Y)
13
14
15 Some data:
16
17 Q P A Y
18 500 105.00 100.00 500.00
19 1800 90.00 195.00 576.54
20 900 75.00 95.00 600.00
21 1400 130.00 145.00 706.23
22 1800 59.00 120.00 750.00
23 1200 114.00 169.00 810.00
24 1300 105.00 95.00 795.00
25 975 165.00 152.00 815.00
26 2000 134.00 202.00 900.00
27
28 A linear fit using the above data
29
30 Q= f + g*P + h*A + i*Y
31
32
33 SUMMARY OUTPUT
34
35 Regression Statistics
36 Multiple R 0.888864171
37 R Square 0.790079515
38 Adjusted R Square 0.664127224
39 Standard Error 283.3424136
40 Observations 9
41
42 ANOVA
43 df SS MS F Significance F
44 Regression 3 1510807.61 503602.5 6.2728475 0.0379091
45 Residual 5 401414.617 80282.92
46 Total 8 1912222.22
47
48 CoefficientsStandard Error t Stat P-value Lower 95%
49 Intercept -142.008306 572.243073 -0.24816 0.8138798 -1613.00355
50 X Variable 1 -9.98905184 3.65624537 -2.73205 0.0411796 -19.3877144
51 X Variable 2 8.198651835 2.7095159 3.025873 0.0292191 1.23363086
52 X Variable 3 1.933355427 0.88602668 2.182051 0.0809034 -0.34424493
53
54 Estimating the results
55 #NAME?
Estimated
56 Q P A Y Q Actual
57 500 105.00 100.00 500.00 596 500
58 1800 90.00 195.00 576.54 1672 1800
59 900 75.00 95.00 600.00 1048 900
60 1400 130.00 145.00 706.23 1114 1400
61 1800 59.00 120.00 750.00 1702 1800
62 1200 114.00 169.00 810.00 1671 1200
63 1300 105.00 95.00 795.00 1125 1300
64 975 165.00 152.00 815.00 1032 975
65 2000 134.00 202.00 900.00 1916 2000
66
67
68 Now try the non-linear fit
69
70 Take the logarithms of the data
71
72 ln(Q) ln(P) ln(A) ln(Y)
73 6.2146 4.6540 4.6052 6.2146
74 7.4955 4.4998 5.2730 6.3570
75 6.8024 4.3175 4.5539 6.3969
76 7.2442 4.8675 4.9767 6.5599
77 7.4955 4.0775 4.7875 6.6201
78 7.0901 4.7362 5.1299 6.6970
79 7.1701 4.6540 4.5539 6.6783
80 6.8824 5.1059 5.0239 6.7032
81 7.6009 4.8978 5.3083 6.8024
82
83 The multiplicative fit
84
85 SUMMARY OUTPUT
86
87 Regression Statistics
88 Multiple R 0.889270086
89 R Square 0.790801287
90 Adjusted R Square 0.665282059
91 Standard Error 0.251221433
92 Observations 9
93
94 ANOVA
95 df SS MS F Significance F
96 Regression 3 1.19286622 0.397622 6.3002402 0.03759558
97 Residual 5 0.31556104 0.063112
98 Total 8 1.50842726
99
100 CoefficientsStandard Error t Stat P-value Lower 95%
101 Intercept -1.81142129 3.01281415 -0.60124 0.573897 -9.55609396
102 X Variable 1 -0.84307889 0.32376373 -2.60399 0.048017 -1.67533869
103 X Variable 2 0.922450092 0.34014869 2.711902 0.0421803 0.04807146
104 X Variable 3 1.266559782 0.51433052 2.462541 0.0570503 -0.05556676
105
106
107
108 Remember that the above results are in logs, so the answer will be the log of sales and you would
109 need to take the anti-log (exp()) of the number ot get the resulting quantity sold
110
111 #NAME?
Estimated Estimated
112 ln(Q) ln(P) ln(A) ln(Y) ln(Q) exp(Q)
113 6.2146 4.6540 4.6052 6.2146 6.3841 592
114 7.4955 4.4998 5.2730 6.3570 7.3105 1496
115 6.8024 4.3175 4.5539 6.3969 6.8514 945
116 7.2442 4.8675 4.9767 6.5599 6.9842 1079
117 7.4955 4.0775 4.7875 6.6201 7.5518 1904
118 7.0901 4.7362 5.1299 6.6970 7.4099 1652
119 7.1701 4.6540 4.5539 6.6783 6.9242 1017
120 6.8824 5.1059 5.0239 6.7032 7.0081 1106
121 7.6009 4.8978 5.3083 6.8024 7.5716 1942
122
123 #NAME?
124
125
H I
gnificance F
Upper 95%
1328.986939
-0.590389268
15.16367281
4.210955787
#NAME?
Error
96 #NAME?
-128
148
-286
-98
471
-175
57
-84
gnificance F
Upper 95%
5.933251388
-0.010819089
1.796828719
2.588686326
of sales and you would
sold
Actual error
500 92 #NAME?
1800 -304
900 45
1400 -321
1800 104
1200 452
1300 -283
975 131
2000 -58
