A seasonal index measures the percentage by which a particular sub-period (month / quarter / season) differs from the annual average due to seasonal effects.
An index of 125 means the period typically exceeds the annual average by 25 %; an index of 80 means it falls 20 % below average.
Seasonal variation can be measured only from data recorded within the year: quarterly, monthly, weekly, daily or hourly. Four methods follow: simple averages, ratio to moving average, ratio to trend, and link relatives.
The simplest method. Assumes no trend.
Quarterly sales (₹ lakhs) for 3 years:
| Year | Q1 | Q2 | Q3 | Q4 |
|---|---|---|---|---|
| 2023 | 60 | 80 | 72 | 68 |
| 2024 | 65 | 88 | 76 | 71 |
| 2025 | 70 | 92 | 80 | 78 |
| Avg \(\bar S_j\) | 65 | 86.67 | 76 | 72.33 |
Overall mean \(\bar S = (65 + 86.67 + 76 + 72.33)/4 = 75\).
Seasonal indices: Q1 = 65/75 × 100 = 86.67; Q2 = 115.56; Q3 = 101.33; Q4 = 96.44. Sum = 400 ✓.
Monthly tourist arrivals show Jan–Dec averages 50, 60, 70, 80, 90, 100, 110, 100, 80, 60, 50, 50. Total = 900, mean = 75. Each month's index = (avg/75)×100.
The textbook states this method's demerit as: “In this method, the trend and cyclic variations can be eliminated … irregular variations also eliminated by averaging … But in practice this is not always true.” The method does nothing to eliminate trend or cycles: it assumes there are none. Averaging the same month over several years removes only the irregular variation.
Why trend matters. If the series rises steadily by \(b\) a month, December is on average \(11b\) higher than January for reasons that have nothing to do with the season, and its simple-average index is inflated accordingly. Exercise 1 below shows it: with a steady upward trend, the indices climb from January to December almost in a straight line. Where there is a trend, use a method that removes it first (§§3–5).
Merits: very simple to understand and calculate; the indices are useful for comparison. Demerit: valid only for a series without trend or cycles, which is why it is seldom used.
Most widely used method when trend is present.
For the data of Example 1 above (60, 80, 72, 68; 65, 88, 76, 71; 70, 92, 80, 78), the 4-quarter totals are 280, 285, 293, 297, 300, 305, 309, 313, 320. Adding them in pairs and dividing by 8 gives centred moving averages from 2023 Q3 to 2025 Q2, and the ratios (actual / centred MA × 100):
| Year | Q1 | Q2 | Q3 | Q4 |
|---|---|---|---|---|
| 2023 | — | — | 101.95 | 94.12 |
| 2024 | 88.14 | 117.92 | 100.50 | 92.51 |
| 2025 | 90.03 | 116.27 | — | — |
Averages: Q1 = 89.08, Q2 = 117.10, Q3 = 101.22, Q4 = 93.31. Sum = 400.72.
Correction factor = 400/400.72 = 0.9982. Adjusted indices: Q1 = 88.92, Q2 = 116.89, Q3 = 101.04, Q4 = 93.15. Sum = 400 ✓.
Monthly sales data: 12-month centred MA computed; ratios formed and averaged month-wise gives 12 seasonal indices summing to 1200 after correction.
A moving average whose period equals the period of an oscillation removes it completely (Unit 1, §4.4). The seasonal pattern has a period of exactly one year, so a centred 12-month (or 4-quarter) moving average contains no seasonal component: under the multiplicative model it estimates \(T \times C\). Dividing,
\[ \frac{y}{\text{centred MA}} \times 100 = \frac{T \times S \times C \times I}{T \times C} \times 100 \] \[ = S \times I \times 100, \]and averaging the ratios for the same month over the years removes \(I\), leaving \(S\). The average may be the mean or, better, the median, which is not pulled by an extreme year.
Merits: the most widely used and flexible method; its indices fluctuate less than ratio-to-trend indices; trend and cycles are both removed. Demerit: no ratios for the first and last six months (two quarters), so not all the data are used.
Used when a clear trend line can be fitted (linear or parabolic).
Quarterly data with fitted linear trend \(T_t = 50 + 2t\). Actual values 60, 80, 72, 68; trend values 52, 54, 56, 58.
Ratios: 60/52 = 115.4; 80/54 = 148.1; 72/56 = 128.6; 68/58 = 117.2.
Average these across all years to get seasonal index for each quarter.
If a study uses parabolic trend \(T_t = a + bt + ct^2\), the ratio-to-trend method still applies — only the trend formula changes.
Fit the trend by least squares to the yearly averages of the quarterly figures (or the yearly totals). If the yearly trend rises by \(b\) a year, it rises by \(b/4\) a quarter.
A yearly average belongs to the middle of the year, between the second and third quarters. So from the yearly trend value \(T\):
\[ \text{Q2} = T - \tfrac{b}{8}, \quad \text{Q3} = T + \tfrac{b}{8}, \quad \text{Q1} = T - \tfrac{3b}{8}, \quad \text{Q4} = T + \tfrac{3b}{8} . \](With \(b = 16\): \(\pm 2\) and \(\pm 6\), as in Worked Problem 3.) Each original value is then expressed as a percentage of its trend value; under the multiplicative model the ratio is \(S \times C \times I\), and averaging by quarter removes \(C\) and \(I\) as far as averaging can.
Merits: uses all the data, with a trend value for every season. Demerit: if there are strong cycles, they are not removed by the trend line and bias the indices more than in the ratio-to-moving-average method.
An alternative method using period-to-period growth ratios.
For 8 quarters (2 years) with values 60, 80, 72, 68, 65, 88, 76, 71:
Link relatives: 80/60 = 133.3; 72/80 = 90.0; 68/72 = 94.4; 65/68 = 95.6; 88/65 = 135.4; 76/88 = 86.4; 71/76 = 93.4.
Averages Q2→Q1 = (133.3 + 135.4)/2 = 134.4; Q3→Q2 = (90 + 86.4)/2 = 88.2; etc. Build chain relatives starting at CR(Q1) = 100.
For monthly data, the method uses 12 link relatives per year; the chain relatives over one cycle should return to 100 (after correction) — any difference is distributed as a "cumulative error" adjustment.
Starting from \(\text{CR}_1 = 100\) and chaining the average link relatives round a full year should bring us back to 100 for the first season, if there were no trend. With an upward trend every link carries a little growth, and the new first-season chain relative,
\[ \text{new CR}_1 = \frac{\overline{\text{LR}}_1 \times \text{CR}_{\text{last}}}{100}, \]comes out above 100. Assuming the trend is linear, the excess has built up evenly over the \(m\) links of the year (\(m = 4\) or 12), so each season carries \(d = (\text{new CR}_1 - 100)/m\) more than the one before. The correction therefore subtracts \(0, d, 2d, \ldots, (m - 1)d\) from the chain relatives of seasons \(1, 2, \ldots, m\). Finally the adjusted chain relatives are scaled so that they total 400 (or 1200), or, equivalently, each is divided by their mean and multiplied by 100.
Merit: uses more of the data than the ratio-to-moving-average method; only one link relative is lost. Demerit: the correction assumes a linear trend. (The textbook says the method works “only if the growth is of constant rate”; the equal steps \(d, 2d, 3d\) are constant absolute steps, which is a linear trend.)
| Method | Assumption | Strength | Weakness |
|---|---|---|---|
| Simple Averages | No trend | Easiest | Biased if trend exists |
| Ratio to Moving Average | Multiplicative model | Most popular & flexible | Loses data at ends |
| Ratio to Trend | Trend can be fitted | Uses all data | Needs a good trend model |
| Link Relatives | Multiplicative | Captures growth pattern | Chain error build-up |
Deseasonalisation means removing the seasonal effect from observed data to reveal the underlying trend-cycle and irregular movement.
Q2 sales = 88 with seasonal index 115.56. Deseasonalised = 88 × 100/115.56 = 76.15. Now comparable across quarters.
Monthly profit 250 with seasonal component 30. Deseasonalised = 250 − 30 = 220. (For low months with negative S, value increases.)
Six problems in the textbook's order, two for each of the simple-averages and ratio-to-moving-average methods and one each for ratio to trend and link relatives, followed by the exercises with their answers checked. Every set of indices is checked against its required total: 400 for quarters, 1200 for months.
Source note. Every figure was recomputed from the data in the problem. The textbook's link-relative solution (Worked Problem 6) is right throughout. Its other solutions each need a correction, given in place: a wrong divisor, an inverted ratio, data copied wrongly into the solution, and in the monthly moving-average problem a run of wrong moving averages.
Obtain seasonal indices by the method of simple averages.
| Year | I | II | III | IV |
|---|---|---|---|---|
| 2003 | 40 | 45 | 36 | 42 |
| 2004 | 50 | 54 | 42 | 48 |
| 2005 | 56 | 58 | 48 | 52 |
| Total | 146 | 157 | 126 | 142 |
| Average | 48.67 | 52.33 | 42.00 | 47.33 |
| Seasonal index | 102.28 | 109.98 | 88.27 | 99.47 |
Each quarter's average is its total over 3 years divided by 3. The average of the four averages is
\[ \bar x = \frac{48.67 + 52.33 + 42 + 47.33}{4} = 47.5833, \]and each index is \(\bar x_i / \bar x \times 100\): 102.28, 109.98, 88.27, 99.47. They total 400, as they must.
Correction. The textbook divides the second-quarter total by 4 instead of 3, giving \(157/4 = 39.25\) instead of \(52.33\). That one slip changes the grand average (to 44.3125) and so every index: it prints 109.83, 88.58, 94.78, 106.81, and turns the strongest quarter, II, into the weakest.
Determine seasonal indices by the method of simple averages.
| Month | 2008 | 2009 | 2010 | Total | Average | Index |
|---|---|---|---|---|---|---|
| Jan | 20 | 38 | 42 | 100 | 33.33 | 102.48 |
| Feb | 24 | 32 | 40 | 96 | 32.00 | 98.38 |
| Mar | 22 | 26 | 38 | 86 | 28.67 | 88.13 |
| Apr | 18 | 32 | 44 | 94 | 31.33 | 96.33 |
| May | 26 | 38 | 46 | 110 | 36.67 | 112.72 |
| Jun | 19 | 26 | 38 | 83 | 27.67 | 85.06 |
| Jul | 32 | 46 | 56 | 134 | 44.67 | 137.32 |
| Aug | 18 | 24 | 30 | 72 | 24.00 | 73.78 |
| Sep | 24 | 28 | 34 | 86 | 28.67 | 88.13 |
| Oct | 26 | 30 | 36 | 92 | 30.67 | 94.28 |
| Nov | 28 | 30 | 40 | 98 | 32.67 | 100.43 |
| Dec | 30 | 40 | 50 | 120 | 40.00 | 122.97 |
The twelve averages total 390.33, so \(\bar x = 390.33/12 = 32.5278\), and each index is the month's average as a percentage of it. July stands out at about 137, August is lowest at about 74.
Note on the data. The problem gives July 2010 as 56; the textbook's solution copies it as 54 (July total 132, average 44), and its indices (102.65, …, 135.51, …) follow from that figure. With the data as given, the indices are the ones above.
Obtain seasonal indices by the ratio-to-trend method.
| Year | I | II | III | IV | Yearly average |
|---|---|---|---|---|---|
| 2005 | 35 | 45 | 41 | 39 | 40 |
| 2006 | 40 | 58 | 54 | 48 | 50 |
| 2007 | 72 | 88 | 64 | 96 | 80 |
| 2008 | 82 | 98 | 80 | 100 | 90 |
| 2009 | 112 | 108 | 86 | 94 | 100 |
Step 1, the yearly trend. With \(x = t - 2007\): \(\sum y = 360\), \(\sum x^2 = 10\), \(\sum xy = -80 - 50 + 0 + 90 + 200 = 160\), so \(a = 360/5 = 72\), \(b = 160/10 = 16\): \(y = 72 + 16(t - 2007)\), with yearly trend values 40, 56, 72, 88, 104.
Step 2, quarterly trend values. The yearly increase 16 is 4 a quarter. The 2005 value 40 sits between Q2 and Q3, so Q2 \(= 40 - 2 = 38\), Q3 \(= 42\), Q1 \(= 34\), Q4 \(= 46\); and so on for each year.
Step 3, data as a percentage of trend (\(S \times C \times I\)):
| Year | Trend values | Data as % of trend | ||||||
|---|---|---|---|---|---|---|---|---|
| I | II | III | IV | I | II | III | IV | |
| 2005 | 34 | 38 | 42 | 46 | 102.9 | 118.4 | 97.6 | 84.8 |
| 2006 | 50 | 54 | 58 | 62 | 80.0 | 107.4 | 93.1 | 77.4 |
| 2007 | 66 | 70 | 74 | 78 | 109.1 | 125.7 | 86.5 | 123.1 |
| 2008 | 82 | 86 | 90 | 94 | 100.0 | 114.0 | 88.9 | 106.4 |
| 2009 | 98 | 102 | 106 | 110 | 114.3 | 105.9 | 81.1 | 85.5 |
Step 4, average and adjust. The quarter totals are 506.3, 571.4, 447.2, 477.1; the averages 101.26, 114.28, 89.45, 95.42, totalling 400.41. Multiplying by \(k = 400/400.41 = 0.9990\) gives the seasonal indices
101.16, 114.16, 89.35, 95.33.
Correction. For 2006 Q1 the textbook prints 125, which is \(50/40\), the trend value divided by the data. The ratio is \(40/50 \times 100 = 80\). With it, the Q1 total is 506.3, not 551.3, and the four averages total about 400.4, not 409.42. The textbook's indices 107.7, 111.7, 87.4, 93.2 come from the inverted ratio; corrected, quarter I is about average, not 8% above it.
Sales (Rs thousand), 1994–97, by quarter: 65, 58, 56, 61; 68, 63, 63, 67; 70, 59, 56, 52; 60, 55, 61, 58. Determine the seasonal variation by the ratio to moving average method.
The 4-quarter totals are 240, 243, 248, 255, 261, 263, 259, 252, 237, 227, 223, 228, 234. Adding them in pairs and dividing by 8 centres them on a quarter:
| Quarter | Sales | Sum of two 4-q totals | Centred MA | Ratio to MA |
|---|---|---|---|---|
| 1994 I | 65 | — | — | — |
| 1994 II | 58 | — | — | — |
| 1994 III | 56 | 483 | 60.375 | 92.75 |
| 1994 IV | 61 | 491 | 61.375 | 99.39 |
| 1995 I | 68 | 503 | 62.875 | 108.15 |
| 1995 II | 63 | 516 | 64.500 | 97.67 |
| 1995 III | 63 | 524 | 65.500 | 96.18 |
| 1995 IV | 67 | 522 | 65.250 | 102.68 |
| 1996 I | 70 | 511 | 63.875 | 109.59 |
| 1996 II | 59 | 489 | 61.125 | 96.52 |
| 1996 III | 56 | 464 | 58.000 | 96.55 |
| 1996 IV | 52 | 450 | 56.250 | 92.44 |
| 1997 I | 60 | 451 | 56.375 | 106.43 |
| 1997 II | 55 | 462 | 57.750 | 95.24 |
| 1997 III | 61 | — | — | — |
| 1997 IV | 58 | — | — | — |
Averaging the three ratios for each quarter gives 108.06, 96.48, 95.16, 98.17, total 397.87; with \(k = 400/397.87 = 1.0054\) the seasonal indices are
108.64, 97.00, 95.67, 98.70.
Note on the data. The problem gives 1997 Q3 as 61; the textbook's solution copies it as 51. That changes the last two moving totals (218 and 224 instead of 228 and 234) and so the 1997 Q1 and Q2 ratios. With 51, the textbook's indices 108.82, 97.89, 95.14, 98.15 are right (its second decimals come from rounding the moving averages to two places).
Monthly sales of a company (Rs lakhs), 2008–10. Determine seasonal variation by the ratio to moving average method.
| Year | Jan | Feb | Mar | Apr | May | Jun |
|---|---|---|---|---|---|---|
| 2008 | 12 | 14 | 15 | 18 | 20 | 16 |
| 2009 | 10 | 17 | 18 | 20 | 28 | 25 |
| 2010 | 13 | 13 | 20 | 22 | 24 | 19 |
| Year | Jul | Aug | Sep | Oct | Nov | Dec |
|---|---|---|---|---|---|---|
| 2008 | 15 | 12 | 18 | 24 | 18 | 14 |
| 2009 | 14 | 20 | 16 | 21 | 23 | 16 |
| 2010 | 16 | 19 | 24 | 18 | 16 | 12 |
The 12-month totals start at \(196\) (all of 2008) and move on by one month at a time: \(196 - 12 + 10 = 194\), \(194 - 14 + 17 = 197\), and so on. Adding consecutive totals in pairs gives the 24-month totals; dividing by 24 gives the centred moving average, which starts at July 2008 and ends at June 2010:
| Month | Sales | 24-month total | Centred MA | Ratio to MA |
|---|---|---|---|---|
| 2008 Jul | 15 | 390 | 16.25 | 92.31 |
| 2008 Aug | 12 | 391 | 16.29 | 73.66 |
| 2008 Sep | 18 | 397 | 16.54 | 108.82 |
| 2008 Oct | 24 | 402 | 16.75 | 143.28 |
| 2008 Nov | 18 | 412 | 17.17 | 104.85 |
| 2008 Dec | 14 | 429 | 17.88 | 78.32 |
| 2009 Jan | 10 | 437 | 18.21 | 54.92 |
| 2009 Feb | 17 | 444 | 18.50 | 91.89 |
| 2009 Mar | 18 | 450 | 18.75 | 96.00 |
| 2009 Apr | 20 | 445 | 18.54 | 107.87 |
| 2009 May | 28 | 447 | 18.62 | 150.34 |
| 2009 Jun | 25 | 454 | 18.92 | 132.16 |
| 2009 Jul | 14 | 459 | 19.12 | 73.20 |
| 2009 Aug | 20 | 458 | 19.08 | 104.80 |
| 2009 Sep | 16 | 456 | 19.00 | 84.21 |
| 2009 Oct | 21 | 460 | 19.17 | 109.57 |
| 2009 Nov | 23 | 458 | 19.08 | 120.52 |
| 2009 Dec | 16 | 448 | 18.67 | 85.71 |
| 2010 Jan | 13 | 444 | 18.50 | 70.27 |
| 2010 Feb | 13 | 445 | 18.54 | 70.11 |
| 2010 Mar | 20 | 452 | 18.83 | 106.19 |
| 2010 Apr | 22 | 457 | 19.04 | 115.54 |
| 2010 May | 24 | 447 | 18.62 | 128.86 |
| 2010 Jun | 19 | 436 | 18.17 | 104.59 |
Arranged by month, two ratios are available for each month; their average is the seasonal index before adjustment:
| Month | 2008 | 2009 | 2010 | Average | Adjusted index |
|---|---|---|---|---|---|
| Jan | — | 54.92 | 70.27 | 62.60 | 62.39 |
| Feb | — | 91.89 | 70.11 | 81.00 | 80.73 |
| Mar | — | 96.00 | 106.19 | 101.10 | 100.76 |
| Apr | — | 107.87 | 115.54 | 111.70 | 111.33 |
| May | — | 150.34 | 128.86 | 139.60 | 139.13 |
| Jun | — | 132.16 | 104.59 | 118.37 | 117.98 |
| Jul | 92.31 | 73.20 | — | 82.76 | 82.48 |
| Aug | 73.66 | 104.80 | — | 89.23 | 88.93 |
| Sep | 108.82 | 84.21 | — | 96.51 | 96.19 |
| Oct | 143.28 | 109.57 | — | 126.42 | 126.00 |
| Nov | 104.85 | 120.52 | — | 112.69 | 112.32 |
| Dec | 78.32 | 85.71 | — | 82.02 | 81.75 |
The twelve averages total 1204.00, so \(k = 1200/1204.00 = 0.9967\), and the adjusted indices total 1200. May (about 139) and October (about 126) are the high months; January (about 62) is the low one.
Correction. The textbook's solution copies February 2010 as 15 (the data give 13), which puts every 12-month total containing that month out by 2, and further slips follow in the later totals (234, 236, 235, 243, … where the data give 221, 223, 222, 230, …). It also prints a moving average of 18.08 for October 2009, where its own 24-month total 464 gives \(464/24 = 19.33\), and a November average of 44.86 for a total of 224.31 (which is \(2 \times 112.16\)). Its adjusted indices, among them 47.94 for November and the factor \(k = 1.0686\), are wrong in consequence.
Obtain seasonal indices by the link relative method.
| Year | I | II | III | IV |
|---|---|---|---|---|
| 2002 | 75 | 60 | 54 | 59 |
| 2003 | 86 | 65 | 63 | 80 |
| 2004 | 90 | 72 | 66 | 85 |
| 2005 | 100 | 78 | 72 | 93 |
Link relatives, each quarter as a percentage of the one before (2003 Q1 is \(86/59 \times 100\), for instance):
| Year | I | II | III | IV |
|---|---|---|---|---|
| 2002 | — | 80.00 | 90.00 | 109.26 |
| 2003 | 145.76 | 75.58 | 96.92 | 126.98 |
| 2004 | 112.50 | 80.00 | 91.67 | 128.79 |
| 2005 | 117.65 | 78.00 | 92.31 | 129.17 |
| Average | 125.303 | 78.395 | 92.725 | 123.55 |
Chain relatives, starting from 100:
\[ \text{CR}_{\text{II}} = \frac{78.395 \times 100}{100} = 78.395, \] \[ \text{CR}_{\text{III}} = \frac{92.725 \times 78.395}{100} = 72.692, \] \[ \text{CR}_{\text{IV}} = \frac{123.55 \times 72.692}{100} = 89.811 . \]Trend correction. Going round once more, \(\text{new CR}_{\text{I}} = 125.303 \times 89.811/100 = 112.536\), not 100, so \(d = (112.536 - 100)/4 = 3.134\), and
\[ \text{I: } 100, \qquad \text{II: } 78.395 - 3.134 = 75.261, \] \[ \text{III: } 72.692 - 2(3.134) = 66.424, \qquad \text{IV: } 89.811 - 3(3.134) = 80.409 . \]Indices. Their mean is \((100 + 75.261 + 66.424 + 80.409)/4 = 80.523\); dividing each by it and multiplying by 100 gives
124.19, 93.47, 82.49, 99.86,
which total 400. The textbook's working here is correct throughout.