There seems to be a widespread impression that Test batting in the third and fourth innings has become easier over time. I myself have held this belief, but when I looked at the numbers, they ran contrary to any preconceived notions we might have had.

The Pitch Quality Index (PQI), one of the most powerful and solid of my derived measures, is an excellent way to determine the true nature of the pitch. Because the expected performances of the players is built in, PQI allows for normalisation across players at the extreme ends of the spectrum. There will be no dumping Don Bradman and Habibul Bashar or Sydney Barnes and Nicky Boje into the same basket.

The PQI is dependent on six values: the expected runs from the batsmen (based on Career-Location Weighted Batting Average), actual runs of the batsmen, expected wickets from the bowlers (based on Career-Location balls per wicket figures), actual wickets of the bowlers, average runs per wicket (RpW) and actual runs-per-wicket values. Actual career figures are used rather than career-to-date figures since this is a secondary value and I wanted some stability in the calculations. All these figures are used to derive the PQI. To start with, the following ratios are determined. Each ratio is capped at 3.0.

- Actual runs to expected runs

- Expected wickets to actual wickets

- Actual RpW to base RpW (details below)

Since each of these PQI ratio values (PQI-R) has a range of 0 to 3.0, the sum of the PQI-R values, which is the first-level PQI, has a range of 0.0 to 9.0. However, since the median of this distribution is around 3.0, a mapping is done to work out a 100-based PQI. The raw PQI of 3.0 maps on to 50.0. The extrapolation is done either side on ranges of 0.0 to 3.0 and 3.0 to 9.0 respectively. A low PQI indicates a bowler-friendly pitch and a high PQI indicates a batting pitch. During the past 147 years, the extreme PQI values are 93.4 (India v New Zealand, Delhi, 1955: NZ 450 for 2d and 112 for 1, India 531 for 7d) and 14.5 (Australia v South Africa, Melbourne, 1932: SA 36 and 45, Australia 153).

My first cut was limited to the match PQI. However, after many requests, consultations and discussions, I have established that the pitch changes significantly during the course of the match and I need to develop individual PQI values for the first and second halves of the match where appropriate. These are called PQI-1, 2 and PQI-3, 4 respectively. However, I will stick to a single PQI-match in the following cases.

- Innings wins

- Draws with three or fewer innings

- Wins/draws with four innings in which fewer than 120 balls were bowled and no wicket was captured in the fourth innings. This is because I need at least one wicket in, say, 15 overs to give me some data to work with

Also, there are these two eminently forgettable Tests where the PQI-match will apply:

- England v Pakistan, The Oval, 2006, which had only three innings since Pakistan forfeited the game.

- South Africa v England, Centurion, 2000 - a travesty of a Test.

An innings win can be a variation of one of two scorelines: 200/400/150 and 400/150/200. In either case, there is a significant change in the first innings itself and this means we cannot get any meaningful data for the third innings. Also, the third innings is the losing innings of one team and a PQI for that alone will be most inaccurate. A drawn match could be any scoreline. Extremes could be 500/100/100 for 8 and 600/900. However, the third innings would be that of one team and no PQI can be developed. In a shortened fourth innings, a score of 33 for no loss in 12 overs does not really provide any information on the second half of the match.

The first Test between New Zealand and West Indies finished just in time for me to include it in the analysis. Conveniently, it went into the single PQI group with a nice middling figure of 52.6. New Zealand's 509 for 7 was compensated by West Indies' cumulative 385 for 20 wickets. So all my analysis results remained intact.

Out of the 2394 Tests, 1485 Tests qualify for calculation of two distinct PQI values. The other 909 Tests have single PQI values. Note the significant drop in the number of Tests with single PQI values in the current period. This is mainly due the propensity of captains to not enforce the follow-on, taking the matches deep into the third and fourth innings, as against the proliferation of three-innings Tests earlier. Australia implemented this no-follow-on policy almost immediately after the 2001 Kolkata Test while the other teams took some time, so there is some drop in the period 2000-2009 and a steep drop in the current decade.

The basis for my analysis is the PQI-Ratio (PQI-R) which is PQI-3, 4/ PQI-1, 2. The highest value for this ratio is 3.322, in Melbourne in 1901-02: PQI-1, 2 = 17.2, PQI-3, 4 = 57.1. The scores were: Australia 112, England 61 in the first innings; Australia 353, England 175 in the second innings. The lowest value is 0.418, at Trent Bridge, 1951: PQI-1, 2 = 75.7, PQI-3, 4 = 31.7. The scores were: South Africa 483 for 9, England 419 for 9 in the first innings. South Africa 121, England 114 in the second innings.

I will do the analysis across all 2394 Tests while splitting the 147 years of Test cricket into six periods: 1877-1939, 1946-1969, 1970-1985, 1986-1999, 2000-2009 and 2010-2020. Henceforth, I will use these period splits as the base for future analysis. I did a major tweak to the PQI processing as I worked on this article. Earlier, I had used the average value of 30.3 as the Base RpW value. This is fine since that is the overall average RpW, however, I realised that the period-level analysis requires a clear tweaking of the Base RpW value since there is considerable variation in RpW averages across the periods. This is illustrated below.

The Average RpW-Match for all four innings for the six periods is 27.1, 29.8, 30.6, 29.7, 32.0 and 31.1. The value across all 2394 Tests is 30.3.

While there were 2394 first innings, 2375 second innings and 2312 third innings, there were only 1611 fourth innings. Hence, using separate RpW values for innings 1/2 and 3/4 is fraught with danger. Over 700 3/4 values will turn out to be third-innings-only values. Hence I have only worked with match-level RpW values. I have used the appropriate period-level average RpW values in all the computations. The changes are very subtle but quite significant.

I will first look at each period and finally look at the summary across all six periods. For each period, I will be presenting an overview visual. This will highlight the Tests with the four highest PQI-R values, the four lowest values and the two values closest to 1.00. Many of these Tests will kindle a memory in readers' minds. The first-innings values - the scores and PQI-1, 2 - are shown on the left and the second-innings values - the scores and PQI-3, 4 - on the right, so that the comparisons for the eight widely varying Tests is visible clearly.

In Abu Dhabi in 2015, Pakistan and England compiled two 500-plus scores in over four days and then collapsed dramatically to lose 14 wickets in 68 overs. The two PQI values were chalk and cheese - 79.4 and 34.0 and a PQI-R of just over 0.4. The Bangladesh-West Indies Mirpur Test of 2012-13 followed the script for the first four days - two 500-plus scores, but made quickly, so that Bangladesh had enough time to collapse in the second innings to lose the Test after scoring 556 in the first. The PQI values are 81.2 and 37.8 and a PQI-R of 0.46.

The middle two Tests have almost identical PQI values, and the PQI-R values are very close to 1.0. The scores make interesting reading.

Moving on to the other side, the first Test in this list is the roller coaster played in Christchurch in 2018. By lunch on the second day, New Zealand and Sri Lanka had finished their first innings for low scores. A three-day finish loomed. Then the recovery started: 820 runs were scored for 14 more wickets. The two PQI values were again the proverbial chalk and cheese - 21.9 and 66.6 and a PQI-R of over 3.0. The next Test featured a Ben Stokes masterclass. The scores tell the story, leading to PQI values of 20.3 and 52.7, and a PQI-R of 2.6. The next two Tests involve Pakistan.

The first three Tests featured in the graphic above - in Adelaide, Port-of-Spain, and Bulawayo - have identical PQI-R values of 0.496. The drop in PQI values is not as drastic as in some cases. The fourth Test featured even has the PQI-R exceeding 0.5.

One of the middle-level Tests had identical PQI values for the two innings groups.

At the SSC in Colombo in 2006, a couple of scores below 200 from Sri Lanka and Pakistan changed dramatically to nearly 800 runs for nine wickets, leading to PQI values of 27.8 and 76.2 and a PQI-R of 2.75. Even if the teams had played for two more days, a decisive result might not have been reached. Four years previously, it was a totally different scoreline in Hamilton. India scored 99 but managed to get a lead of five runs. Then they improved to 154 but lost the match by four wickets. Despite the low scores, the PQI values were 15.4 and 33.7 leading to a PQI-R of 2.2. The match PQI was a low 20.6.

The two Tests featured on the PQI-dropping scenario - England v Pakistan, Edgbaston, 1987 and West Indies v New Zealand, Antigua, 1995-96 - had PQI-R values either side of 0.5, indicating that batting in the second innings became progressively tougher. England just about held on at Edgbaston.

At the other end of the spectrum was the India-West Indies Delhi Test in 1987-88. India were shot out for 75 in less than two and a half hours of play. They struck back, dismissing the strong Viv Richards-led West Indies for 127. Then a Dilip Vengsarkar hundred helped India to a respectable 327, only for a Richards masterclass to help West Indies win by five wickets. The relevant values were 17.6, 55.5 and 3.16. India recovered brilliantly against New Zealand in Mohali in 1999-00 and almost won the Test.

The four Tests in this period where batting became tougher in the second innings - in the Pakistan-England Karachi Test in 1972-73, India-West Indies Mumbai Test in 1974-75, New Zealand-England Christchurch in 1977-78, and Australia-Pakistan Melbourne Test in 1972-73 - all have PQI-R values exceeding 0.5.

The middle Tests provide contrasting scorelines.

The first Test featured where batting became easier in the second innings is the Centenary Test in Melbourne in 1977. Two scores either side of 100 were followed by two 400-plus totals. Derek Randall's magnificent 174 could not prevent a repeat of the 1877 win margin of 45 runs. The relevant numbers were 21.3, 58.8 and 2.75. Four years later, in Perth, Australia improved dramatically in the second innings and Pakistan less effectively, ending in a huge Pakistan loss.

In the Trent Bridge Test in 1951, South Africa and England crossed 400 and then plummeted to just over 100 in the third and fourth innings. The PQI started at 75.7 and dropped to 31.7, resulting in a PQI-R of 0.418.

In Adelaide later that year, two strong batting teams folded for a combined total of 187 before stumps were drawn on the first day. To boot, another two wickets of the third innings fell on the opening day. Then Australia rallied to score 255 and West Indies won comfortably by six wickets. The key PQI values were 15.1, 49.9 and 3.323. This is the highest PQI-R ever. A month later, in the same series, at the SCG, the two first innings produced almost identical scores. Then Australia scored 377 and won by a big margin. The numbers were 15.5, 45.3 and 2.932.

In the 1934 Ashes Test at The Oval, the PQI-R slumped to 0.537 despite a third innings of 327, mainly because of the first innings of 701. The two PQI values were 69.4 and 37.2.

In the MCG Ashes Test in 1901-02, the scores of 112 and 61 resulted in completion of the first two innings before tea. Then Australia lost five wickets for 48. Next day, Australia progressed to 353 and won by well over 200 runs. The relevant figures worked out to 17.2, 57.1 and 3.322. This match is a third decimal point away from the highest value. The Melbourne Test of 1894-95 was similar in many ways. A first-innings aggregate of 198 became an aggregate of over 800. The PQI values were 24.1, 71.1 and 2.945.

All four of the low PQI-R values are lower than 0.5. The most significant fact to be noted is that three of the four Tests are from the current period. This will lend credence to the inescapable conclusion that batting has indeed become more and more difficult as we have moved on to the current period in Tests.

The two Tests with similar first and second-innings numbers have identical PQI-1, 2 and PQI-3, 4 values.

All four of the high PQI-R values are greater than 3.0. The matches are distributed across four periods. The first two Tests have almost identical PQI-R values.

First, let me get all the numbers cleared up. The light grey band represents the matches in which the pitch conditions improved significantly as the match progressed. The PQI-3, 4 values are over 1.1 times the PQI-1, 2 values. The grey band represents the matches in which the PQI-1, 2 and PQI-3, 4 values are within 10% of each other. In other words, the pitch remained almost the same throughout. The blue band represents the matches in which the pitch conditions worsened significantly as the match progressed. The PQI-3, 4 values are below 0.9 of the PQI-1, 2 values. The percentage value below the period description represents the ratio between the PQI-R>1.1 matches and PQI-R<0.9 matches.

There is very clear indication that our initial premise is not correct. Nowadays, the pitches do not improve significantly as the match progresses. In the period of 2010-20, in only 83 Tests did the PQI-3, 4 improve significantly, while in 130 Tests the PQI-3, 4 reduced significantly. This was the case in the previous period also - 104 against 137.

However, as we go back, the situation starts changing. In the 1986-1999 period, the numbers are closer together - 92 against 107. In the '70s and the early part of the '80s, we have 71 against 79. However, as we move into the 1950s and 1960s, there is a clear shift. In 95 Tests, the PQI-R>1.1 Tests were more than the PQI-R<0.9 Tests - 95 against 75. In the first fifty years, this proportion remained likewise - 67 against 51.

It should be understood that these are based on the actual numbers vs expected values (reflecting the player figures) and the RpW for the periods. In other words, the numbers themselves are not comparable but the PQI-R values are.

In the period between 1877 and 1939, the ratio of Tests in which the PQI-R was greater than 1.1, compared to the matches in which it was less than 0.9, was 1.314 (67 Tests to 51). In the last decade, that ratio has reduced to 0.638 (83 Tests to 130). Why would the current periods have such significant variations? A few reasons come to mind:

1. The significantly higher RpW values - 32.1 and 31.1 against the late-20s values in earlier periods.

2. The non-enforcement of the follow-on means that the leading team itself goes for runs usually ending with scores of 200 for 6 or so and the last batting team invariably has to bat in difficult conditions.

3. The Expected Runs for the second innings remain at somewhat higher levels.

4. The presence of quite a few teams that are finding their feet in the Test arena. Three teams have made their Test debuts in the last two decades.

5. Possibly more attacking batting and bowling as the match progresses. Teams are not allowed to go defensive.

The final nail on the coffin, so to speak, is a comparison between the Means of the PQI-R values across the periods.

1877-1939: 1.11

1946-1969: 1.06

1970-1985: 1.06

1986-1999: 1.04

2000-2009: 1.02

2010-2020: 0.98

A very revealing set of numbers, indeed. The current decade is the only one in which the Mean for PQI-R is below 1.0. It is at 0.98, which is 0.06 below the all-time Mean value of 1.04. Whatever be the reasons, it is clear that the teams are not finding batting easier as the match progresses. If anything, batting has become decidedly more difficult. The gap between PQI-1, 2 and PQI-3, 4 is widening.

*Email me with your comments and I will respond. This email id is to be used only for sending in comments.*

Anantha Narayanan has written for ESPNcricinfo and CastrolCricket and worked with a number of companies on their cricket performance ratings-related systems