Time Series and Forecasting

CHAPTER 18

TIME SERIES AND FORECASTING

1. for 2003, t = 8

= 52.4 + 30.6t = 52.4 + 30.6(8) = 297.2

3.

= 1.30 + 0.90t = 1.30 + 0.90(7) = 7.6(tonnes)

5. (a)YearCode (t)Sales (Y)ln (Y)t*ln(Y)t2

199611.10.095310.095311

199721.50.4054650.810934

1998320.6931472.0794429

199942.40.8754693.50187516

200053.11.1314025.65701125

Totals1510.13.20079312.1445755

Note: “ln” here stands for natural logarithms (pronounced as lan of..).

ln (b) = (12.72285) / (50) = 0.2545

ln (a) = - 0.1235

OR .

The logarithmic equation therefore is:

Note: gives a = 0.884; b = 1.289

And, hence the equation in exponential form as;

(b)The average percentage growth rate is found as: (b-1)*100 = (1.289-1)*100 = 28.9%.

(c) Sales for the year 2003 are found as sales for t = 8

For t = 8: = - 0.1235 + 0.2545 (8) = 1.9125. Since, , = 6.769.

Minitab output for parts a, b and c is given below:

Trend Analysis (Exponential)

Data Sales

Length 5.00000

NMissing 0

Fitted Trend Equation

Yt = 0.884708*(1.28945**t)

Accuracy Measures

MAPE: 2.88916

MAD: 0.0545174

MSD: 0.00362965

Row Period FORE3

1 6 4.06663

2 7 5.24373

3 8 6.76155

(d)Quadratic (parabola) trend provided a better fit.

This can be judged in two ways. (1) Graphically the fitted points are closer to the actual points.

(2) Accuracy measures such as MAPE, MAD or MSD that measure the errors, on average, between the actual and the fitted values. A lower value of these measures indicates a better fit. All these measures have smaller values for quadratic trend compared to the exponential trend as shown below by the Minitab output.

Trend Analysis (Exponential with accurate data)

Data Sales

Length 5.00000

NMissing 0

Fitted Trend Equation

Yt = 1.03385*(1.19283**t)

Accuracy Measures

MAPE: 11.5974

MAD: 0.222229

MSD: 0.0660617

Trend Analysis (Quadratic with accurate data)

Data Sales

Length 5.00000

NMissing 0

Fitted Trend Equation

Yt = 0.2 + 0.932857*t - 0.107143*t**2

Accuracy Measures

MAPE: 5.86221

MAD: 0.102857

MSD: 0.0132571

7.QuarterSeasonal

Indexes

10.690

21.666

31.168

40.476

MegaStat Output

Centered Moving Average and Deseasonalization
Centered
Moving / Ratio to / Seasonal / Absences
t / Year / Quarter / Absences / Average / CMA / Indexes / Deseasonalized
1 / 1 / 1 / 4 / 0.690 / 5.8
2 / 1 / 2 / 10 / 1.666 / 6.0
3 / 1 / 3 / 7 / 6.125 / 1.143 / 1.168 / 6.0
4 / 1 / 4 / 3 / 6.500 / 0.462 / 0.476 / 6.3
5 / 2 / 1 / 5 / 7.000 / 0.714 / 0.690 / 7.2
6 / 2 / 2 / 12 / 7.375 / 1.627 / 1.666 / 7.2
7 / 2 / 3 / 9 / 7.625 / 1.180 / 1.168 / 7.7
8 / 2 / 4 / 4 / 8.250 / 0.485 / 0.476 / 8.4
9 / 3 / 1 / 6 / 9.125 / 0.658 / 0.690 / 8.7
10 / 3 / 2 / 16 / 9.500 / 1.684 / 1.666 / 9.6
11 / 3 / 3 / 12 / 1.168 / 10.3
12 / 3 / 4 / 4 / 0.476 / 8.4
Calculation of Seasonal Indexes
1 / 2 / 3 / 4
1 / 1.143 / 0.462
2 / 0.714 / 1.627 / 1.180 / 0.485
3 / 0.658 / 1.684
mean: / 0.686 / 1.656 / 1.162 / 0.473 / 3.976
adjusted: / 0.690 / 1.666 / 1.168 / 0.476 / 4.000

9.estimated pairsQuarterly forecast

t(millions)Seasonal index(%)(millions)

2140.05110.044.055

2241.80120.050.160

2343.5580.034.840

2445.3090.040.770

11.= 5.5528 + 0.3787 (t). The following are the sales estimates.

EstimateIndexForecast

10.0970.6906.967

10.4761.66617.452

10.8541.16812.678

11.2330.4765.347

Linear Trend on Deseasonalized Absences: MegaStat Output

Regression Analysis
0.855 / r²
0.925 / r
0.590 / std. error of estimate
12 / observations
1 / predictor variable
Desea-Abs / dependent variable
confidence interval
variables / coefficients / std. error / t (df=10) / p-value / 95% lower / 95% upper
intercept / a = / 5.5528
t / b = / 0.37867852 / 0.04937152 / 7.67 / 1.70E-05 / 0.26867190 / 0.48868515
ANOVA table
Source / SS / df / MS / F / p-value
Regression / 20.5058 / 1 / 20.5058 / 58.83 / 1.70E-05
Residual / 3.4857 / 10 / 0.3486
Total / 23.9915 / 11
Predicted values for: Desea-Abs
95% Confidence Interval / 95% Prediction Interval
t / Predicted / lower / upper / lower / upper
12 / 10.09697 / 9.28734 / 10.90659 / 8.55230 / 11.64164
13 / 10.47565 / 9.56740 / 11.38390 / 8.87708 / 12.07422
14 / 10.85433 / 9.84510 / 11.86355 / 9.19630 / 12.51235
15 / 11.23300 / 10.12108 / 12.34492 / 9.51054 / 12.95547

13.a.A decline, as shown by the linear trend.

b.Least square is preferable for estimating long-term trend. Least square method irons out all fluctuations including seasonal, cyclical and irregular. However, since the least squares is influenced by unusual observations, a better method to estimate long term trend would consist of using least square method on the deseasonalized series obtained by an application of the moving average method on the raw data.

c.11200, approximately.

15.a.

b.= 1.00455 + 0.04409t, using t = 1 for 1990

c.for 1993, = 1.18091, and for 1998 = 1.40136

d.for 2005, = 1.70999

e.Each asset, on average, turned over 0.044 times

17.a.

b.= 49.140 – 2.9829t

c.for 1997, = 40.1913 and for 1999, = 34.2255

d.for 2003 = 22.2939

e.The number of employees decreases, on average, at a rate of 2983 per year.

19.a.

b.For t = 4: = 2.8264;

=16.8846

For t = 9: = 4.1749

=65.033

c. 29.92%, which is the Inv (ln) of 0.2617 minus 1 (in percent form)

d.for 2002, t = 12 and = 4.96

= 142.59

Computer output based on MegaStat (Excel) is given below. For manual calculations, see question 5.

MegaStat Output

Regression Analysis
1.000 / r²
1.000 / r
0.009 / std. error of estimate
11 / observations
1 / predictor variable
ln(Sales) / dependent variable
confidence interval
variables / coefficients / std. error / t (df=9) / p-value / 95% lower / 95% upper
intercept / a = / 1.8196
t / b = / 0.2617 / 0.00090 / 290.41 / 3.46E-19 / 0.2597 / 0.2638
coefficients in terms of the model: abx
6.169 / = a, beginning value
1.299 / = b, growth factor
29.92% / average rate of change
ANOVA table
Source / SS / df / MS / F / p-value
Regression / 7.5354 / 1 / 7.5354 / 84340.67 / 3.46E-19
Residual / 0.0008 / 9 / 0.0001
Total / 7.5362 / 10
Predicted values for: ln(Sales)
95% Confidence Interval / 95% Prediction Interval
t / Predicted / lower / upper / lower / upper
1 / 2.08131 / 2.06925 / 2.09337 / 2.05676 / 2.10586
2 / 2.34304 / 2.33264 / 2.35343 / 2.31926 / 2.36681
3 / 2.60477 / 2.59589 / 2.61366 / 2.58162 / 2.62793
4 / 2.86650 / 2.85888 / 2.87413 / 2.84380 / 2.88921
5 / 3.12824 / 3.12148 / 3.13500 / 3.10581 / 3.15066
6 / 3.38997 / 3.38352 / 3.39642 / 3.36764 / 3.41230
7 / 3.65170 / 3.64494 / 3.65846 / 3.62928 / 3.67413
8 / 3.91343 / 3.90581 / 3.92106 / 3.89073 / 3.93614
9 / 4.17517 / 4.16628 / 4.18405 / 4.15201 / 4.19832
10 / 4.43690 / 4.42650 / 4.44730 / 4.41312 / 4.46068
11 / 4.69863 / 4.68657 / 4.71069 / 4.67408 / 4.72318
12 / 4.96036 / 4.94654 / 4.97419 / 4.93490 / 4.98583
Predicted values for: Sales
95% Confidence Interval / 95% Prediction Interval
t / Predicted / lower / upper / lower / upper
1 / 8.01 / 7.92 / 8.11 / 7.82 / 8.21
2 / 10.41 / 10.31 / 10.52 / 10.17 / 10.66
3 / 13.53 / 13.41 / 13.65 / 13.22 / 13.85
4 / 17.58 / 17.44 / 17.71 / 17.18 / 17.98
5 / 22.83 / 22.68 / 22.99 / 22.33 / 23.35
6 / 29.67 / 29.47 / 29.86 / 29.01 / 30.34
7 / 38.54 / 38.28 / 38.80 / 37.69 / 39.41
8 / 50.07 / 49.69 / 50.45 / 48.95 / 51.22
9 / 65.05 / 64.48 / 65.63 / 63.56 / 66.57
10 / 84.51 / 83.64 / 85.40 / 82.53 / 86.55
11 / 109.80 / 108.48 / 111.13 / 107.13 / 112.53
12 / 142.65 / 140.69 / 144.63 / 139.06 / 146.32
Data:
t / ln(Sales) / Sales
1 / 2.07944 / 8.0
2 / 2.34181 / 10.4
3 / 2.60269 / 13.5
4 / 2.86790 / 17.6
5 / 3.12676 / 22.8
6 / 3.37759 / 29.3
7 / 3.67377 / 39.4
8 / 3.92197 / 50.5
9 / 4.17439 / 65.0
10 / 4.43201 / 84.1
11 / 4.69135 / 109.0

21.a.

b.Linear Trend:

Logarithmic Trend:

Logarithmic trend is more accurate.

Accuracy measures are given by Minitab in terms of MAPE (mean absolute percentage error), MAD (mean absolute deviation) and MSD (mean squared deviation). Lower the values of these measures, better the fit. They are all lower for logarithmic trend compared to linear trend. MegaStat gives R2. However, the values of R2 for two equations are not comparable since the dependent variable are expressed in different units.

c.For1993: t = 4:

Linear Trend: = 4164.6

Logarithmic Trend: = 8.3096

= 4062.69

For1998: t = 9:

Linear Trend: = 6901.73

Logarithmic Trend: = 8.8031

= 6654.84

d.For 2003, t = 14

Linear Trend: = 9638.18

Logarithmic Trend: = 9.2966

= 10900.89

e. Based on the Logarithmic trend: 10.37%, which is the Inv (ln) of 0.0987 minus 1 (in percent form)

Note: Answers may differ from computer output due to rounding.

MegaStat (Excel)

Regression Analysis / (LINEAR)
0.894 / r²
0.946 / r
711.281 / std. error of estimate
12 / observations
1 / predictor variable
TSE / dependent variable
confidence interval
variables / coefficients / std. error / t (df=10) / p-value / 95% lower / 95% upper
intercept / a = / 1,976.1165
t / b = / 547.292203 / 59.480278 / 9.20 / 3.39E-06 / 414.761861 / 679.822545
ANOVA table
Source / SS / df / MS / F / p-value
Regression / ########## / 1 / ########## / 84.66 / 3.39E-06
Residual / ########## / 10 / ##########
Total / ########## / 11
Predicted values for: TSE
95% Confidence Interval / 95% Prediction Interval
t / Predicted / lower / upper / lower / upper
1 / 2,523.409 / 1,662.811 / 3,384.006 / 719.989 / #######
2 / 3,070.701 / 2,319.046 / 3,822.356 / 1,316.655 / #######
3 / 3,617.993 / 2,966.479 / 4,269.507 / 1,904.469 / #######
4 / 4,165.285 / 3,600.410 / 4,730.161 / 2,482.794 / #######
5 / 4,712.578 / 4,213.751 / 5,211.404 / 3,051.096 / #######
6 / 5,259.870 / 4,797.594 / 5,722.145 / 3,608.993 / #######
7 / 5,807.162 / 5,344.886 / 6,269.438 / 4,156.286 / #######
8 / 6,354.454 / 5,855.628 / 6,853.280 / 4,692.973 / #######
9 / 6,901.746 / 6,336.871 / 7,466.622 / 5,219.255 / #######
10 / 7,449.039 / 6,797.525 / 8,100.552 / 5,735.515 / #######
11 / 7,996.331 / 7,244.676 / 8,747.985 / 6,242.284 / #######
12 / 8,543.623 / 7,683.026 / 9,404.220 / 6,740.204 / #######
13 / 9,090.915 / 8,115.518 / 10,066.312 / 7,229.976 / #######
14 / 9,638.207 / 8,543.996 / 10,732.419 / 7,712.333 / #######
Data:
t / TSE
1 / 3,421.10
2 / 3,469.48
3 / 3,402.92
4 / 3,904.22
5 / 4,284.06
6 / 4,433.88
7 / 5,268.01
8 / 6,458.20
9 / 6,757.27
10 / 7,059.11
11 / 9,607.74
12 / 8,336.20
Regression Analysis / (LOGARITHMIC)
0.945 / r²
0.972 / r
0.090 / std. error of estimate
12 / observations
1 / predictor variable
ln(TSE) / dependent variable
confidence interval
variables / coefficients / std. error / t (df=10) / p-value / 95% lower / 95% upper
intercept / a = / 7.9148
t / b = / 0.0987 / 0.00751 / 13.14 / 1.24E-07 / 0.0819 / 0.1154
coefficients in terms of the model: abx
2737.376 / = a, beginning value
1.104 / = b, growth factor
10.37% / average rate of change
ANOVA table
Source / SS / df / MS / F / p-value
Regression / 1.3921 / 1 / 1.3921 / 172.66 / 1.24E-07
Residual / 0.0806 / 10 / 0.0081
Total / 1.4727 / 11
Predicted values for: ln(TSE)
95% Confidence Interval / 95% Prediction Interval
t / Predicted / lower / upper / lower / upper
1 / 8.01342 / 7.90478 / 8.12206 / 7.78576 / 8.24108
2 / 8.11209 / 8.01720 / 8.20698 / 7.89066 / 8.33352
3 / 8.21075 / 8.12851 / 8.29300 / 7.99444 / 8.42707
4 / 8.30942 / 8.23811 / 8.38073 / 8.09702 / 8.52182
5 / 8.40809 / 8.34511 / 8.47106 / 8.19834 / 8.61783
6 / 8.50675 / 8.44839 / 8.56511 / 8.29835 / 8.71516
7 / 8.60542 / 8.54706 / 8.66377 / 8.39701 / 8.81382
8 / 8.70408 / 8.64111 / 8.76706 / 8.49434 / 8.91383
9 / 8.80275 / 8.73144 / 8.87406 / 8.59035 / 9.01515
10 / 8.90142 / 8.81917 / 8.98366 / 8.68510 / 9.11773
11 / 9.00008 / 8.90519 / 9.09497 / 8.77865 / 9.22151
12 / 9.09875 / 8.99011 / 9.20739 / 8.87109 / 9.32641
13 / 9.19741 / 9.07428 / 9.32055 / 8.96249 / 9.43234
14 / 9.29608 / 9.15795 / 9.43421 / 9.05296 / 9.53920
Predicted values for: TSE
95% Confidence Interval / 95% Prediction Interval
t / Predicted / lower / upper / lower / upper
1 / 3021.24 / 2710.21 / 3367.96 / 2406.09 / 3793.65
2 / 3334.53 / 3032.67 / 3666.44 / 2672.20 / 4161.02
3 / 3680.31 / 3389.73 / 3995.80 / 2964.43 / 4569.08
4 / 4061.95 / 3782.39 / 4362.19 / 3284.68 / 5023.16
5 / 4483.17 / 4209.56 / 4774.56 / 3634.92 / 5529.37
6 / 4948.06 / 4667.57 / 5245.41 / 4017.23 / 6094.59
7 / 5461.17 / 5151.59 / 5789.35 / 4433.80 / 6726.58
8 / 6027.48 / 5659.62 / 6419.24 / 4887.03 / 7434.06
9 / 6652.51 / 6194.65 / 7144.22 / 5379.52 / 8226.75
10 / 7342.36 / 6762.65 / 7971.77 / 5914.15 / 9115.48
11 / 8103.75 / 7370.16 / 8910.36 / 6494.12 / 10112.33
12 / 8944.09 / 8023.32 / 9970.53 / 7123.02 / 11230.74
13 / 9871.57 / 8727.91 / 11165.09 / 7804.78 / 12485.67
14 / 10895.23 / 9489.57 / 12509.11 / 8543.79 / 13893.84
Data:
t / ln(TSE) / TSE
1 / 8.137717 / 3,421.10
2 / 8.151760 / 3,469.48
3 / 8.132389 / 3,402.92
4 / 8.269813 / 3,904.22
5 / 8.362656 / 4,284.06
6 / 8.397030 / 4,433.88
7 / 8.569408 / 5,268.01
8 / 8.773106 / 6,458.20
9 / 8.818374 / 6,757.27
10 / 8.862074 / 7,059.11
11 / 9.170324 / 9,607.74
12 / 9.028363 / 8,336.20

Minitab Output

Trend Analysis (Linear)

Data TSE

Length 12.0000

NMissing 0

Fitted Trend Equation

Yt = 1976.12 + 547.292*t

Accuracy Measures

MAPE: 9.84461

MAD: 501.938

MSD: 421600

Row Period FORE1

1 13 9090.92

2 14 9638.21

Trend Analysis (Exponential/Logarithmic)

Data TSE

Length 12.0000

NMissing 0

Fitted Trend Equation

Yt = 2737.38*(1.10370**t)

Accuracy Measures

MAPE: 6.90491

MAD: 400.584

MSD: 294126

Row Period FORE2

1 13 9871.6

2 14 10895.2

23.a.July 87.5, August 92.9, September 99.3, October 109.1 (all in %)

b.MonthTotalMeanSeasonal

July348.987.22586.777

Aug.368.192.02591.552

Sept.395.098.75098.242

Oct.420.4105.100104.560

Nov.496.2124.050123.412

Dec.572.3143.075142.340

Jan.333.583.37582.946

Feb.297.574.37573.993

March347.386.82586.379

April481.3120.325119.707

May396.299.05098.541

June368.192.02591.552

1206.200

Correction = 1200/1206.2 = 0.99486

c.April, November, and December are periods of high sales, while February is low.

25.a.Seasonal Index by Quarter

Average SISeasonal

QuarterComponentIndex

10.50140.5027

21.09091.0936

31.77091.7753

40.63540.6370

b.The production is the largest in the third quarter. It is 77.5% above the average quarter. The second quarter is also above average. The first and fourth quarters are well below average, with the first quarter at about 50% of a typical quarter.

27.a. Seasonal Index by Quarter

Average SISeasonal

QuarterComponentIndex

10.55490.5577

20.82540.8296

31.51021.5178

41.09731.1029

b.= 7.667 + 0.0023t

c.PeriodProductionIndexForecast

217.71530.55774.3028

227.71760.82966.4025

237.71991.517811.7173

247.72221.10298.5168

29. Seasonal Index by Quarter

Average SISeasonal

QuarterComponentIndex

11.19621.2053

21.01351.0212

30.62530.6301

41.13711.1457

The regression equation is: = 43.611 + 7.21153t

PeriodVisitorsIndexForecast

29252.861.2053304.77

30260.071.0212265.58

31267.290.6301168.42

32274.501.1457314.50

In 1997 there were a total of 928 visitors. A ten percent increase in 1998 means there will be 1021 visitors. The quarterly estimates are 1021/4 = 255.25 visitors per quarter.

PeriodVisitorsIndexForecast

Winter255.251.2053307.65

Spring255.251.0212260.66

Summer255.250.6301160.83

Fall255.251.1457292.44

The regression approach is probably superior because the trend is considered.

31.a. Q1Q2Q3Q4

86.4104.7101.6107.4

b. Sales are higher in quarter 2 and 4 and lower in quarters 1 and 3.

c. Estimated Sales on deseasonalized data: = 53.252.7+ 843.11 (t)

d. tForecast

21 70,958.05024

22 71,801.16235

23 72,644.27445

24 73,487.38656

The Excel (MegaStat) out put is given below.

Centered Moving Average and Deseasonalization
Centered
Moving / Ratio to / Seasonal / Sales($M)
t / Year / Quarter / Sales($M) / Average / CMA / Indexes / Deseasonalized
1 / 1996 / 1 / 47,287 / 0.864 / 54,743.0
2 / 1996 / 2 / 57,047 / 1.047 / 54,486.0
3 / 1996 / 3 / 55,731 / 55520.250 / 1.004 / 1.016 / 54,871.4
4 / 1996 / 4 / 60,805 / 56461.500 / 1.077 / 1.074 / 56,640.2
5 / 1997 / 1 / 49,709 / 57711.500 / 0.861 / 0.864 / 57,546.9
6 / 1997 / 2 / 62,155 / 58891.000 / 1.055 / 1.047 / 59,364.7
7 / 1997 / 3 / 60,623 / 59774.750 / 1.014 / 1.016 / 59,687.9
8 / 1997 / 4 / 65,349 / 60420.000 / 1.082 / 1.074 / 60,872.9
9 / 1998 / 1 / 52,235 / 61043.625 / 0.856 / 0.864 / 60,471.2
10 / 1998 / 2 / 64,791 / 61503.375 / 1.053 / 1.047 / 61,882.4
11 / 1998 / 3 / 62,976 / 61995.000 / 1.016 / 1.016 / 62,004.6
12 / 1998 / 4 / 66,674 / 62612.875 / 1.065 / 1.074 / 62,107.2
13 / 1999 / 1 / 54,843 / 63464.750 / 0.864 / 0.864 / 63,490.4
14 / 1999 / 2 / 67,126 / 64609.875 / 1.039 / 1.047 / 64,112.6
15 / 1999 / 3 / 67,456 / 65738.625 / 1.026 / 1.016 / 66,415.5
16 / 1999 / 4 / 71,355 / 66820.000 / 1.068 / 1.074 / 66,467.6
17 / 2000 / 1 / 59,192 / 67904.625 / 0.872 / 0.864 / 68,525.1
18 / 2000 / 2 / 71,428 / 68854.875 / 1.037 / 1.047 / 68,221.5
19 / 2000 / 3 / 71,831 / 1.016 / 70,723.0
20 / 2000 / 4 / 74,582 / 1.074 / 69,473.5
Calculation of Seasonal Indexes
1 / 2 / 3 / 4
1996 / 1.004 / 1.077
1997 / 0.861 / 1.055 / 1.014 / 1.082
1998 / 0.856 / 1.053 / 1.016 / 1.065
1999 / 0.864 / 1.039 / 1.026 / 1.068
2000 / 0.872 / 1.037
mean: / 0.863 / 1.046 / 1.015 / 1.073 / 3.997
adjusted: / 0.864 / 1.047 / 1.016 / 1.074 / 4.000
Regression Analysis
0.978 / r²
0.989 / r
776.800 / std. error of estimate
20 / observations
1 / predictor variable
Des-Sales($M) / dependent variable
variables / coefficients / std. error / t (df=18) / p-value
intercept / a = / 53,252.6961
t / b = / 843.11210386 / 30.12300672 / 27.99 / 2.73E-16
ANOVA table
Source / SS / df / MS / F / p-value
Regression / 472,707,283.0830 / 1 / 472,707,283.0830 / 783.38 / 2.73E-16
Residual / 10,861,524.5394 / 18 / 603,418.0300
Total / 483,568,807.6223 / 19
Predicted values for: Des-Sales($M)
95% Confidence Interval / 95% Prediction Interval
t / Predicted / lower / upper / lower / upper
1 / 54,095.80817 / 53,392.50581 / 54,799.11052 / 52,318.71809 / 55,872.89824
2 / 54,938.92027 / 54,288.88814 / 55,588.95240 / 53,182.23114 / 56,695.60940
3 / 55,782.03237 / 55,183.31742 / 56,380.74733 / 54,043.67840 / 57,520.38635
4 / 56,625.14448 / 56,075.24657 / 57,175.04239 / 54,902.99389 / 58,347.29506
5 / 57,468.25658 / 56,963.94905 / 57,972.56411 / 55,760.11695 / 59,176.39622
6 / 58,311.36869 / 57,848.47050 / 58,774.26688 / 56,614.99324 / 60,007.74413
7 / 59,154.48079 / 58,727.59245 / 59,581.36913 / 57,467.57576 / 60,841.38582
8 / 59,997.59289 / 59,599.84569 / 60,395.34010 / 58,317.82572 / 61,677.36007
9 / 60,840.70500 / 60,463.63431 / 61,217.77568 / 59,165.71329 / 62,515.69671
10 / 61,683.81710 / 61,317.52210 / 62,050.11211 / 60,011.21824 / 63,356.41596
11 / 62,526.92921 / 62,160.63420 / 62,893.22421 / 60,854.33034 / 64,199.52807
12 / 63,370.04131 / 62,992.97063 / 63,747.11199 / 61,695.04960 / 65,045.03302
13 / 64,213.15341 / 63,815.40621 / 64,610.90062 / 62,533.38624 / 65,892.92059
14 / 65,056.26552 / 64,629.37718 / 65,483.15385 / 63,369.36049 / 66,743.17055
15 / 65,899.37762 / 65,436.47943 / 66,362.27581 / 64,203.00217 / 67,595.75307
16 / 66,742.48973 / 66,238.18219 / 67,246.79726 / 65,034.35009 / 68,450.62936
17 / 67,585.60183 / 67,035.70392 / 68,135.49974 / 65,863.45124 / 69,307.75241
18 / 68,428.71393 / 67,829.99897 / 69,027.42889 / 66,690.35996 / 70,167.06790
19 / 69,271.82604 / 68,621.79391 / 69,921.85816 / 67,515.13691 / 71,028.51516
20 / 70,114.93814 / 69,411.63579 / 70,818.24049 / 68,337.84807 / 71,892.02821
21 / 70,958.05024 / 70,199.93620 / 71,716.16429 / 69,158.56370 / 72,757.53679
22 / 71,801.16235 / 70,987.00642 / 72,615.31828 / 69,977.35731 / 73,624.96738
23 / 72,644.27445 / 71,773.08382 / 73,515.46509 / 70,794.30471 / 74,494.24420
24 / 73,487.38656 / 72,558.35123 / 74,416.42188 / 71,609.48305 / 75,365.29006

33.Answers will vary.

35.

The median is increasing $18,150 per year, while the average is increasing much more rapidly ($70,000 per year).

18-1