The Aging Athlete (6) – Men’s 1500m

Photo Courtesy Doug Shaggy Smith

Based upon the previous post, we have learned that 1500m Male runners’ performance after age 35, on average, tends to decline about 1/2% per year until M50. Then the erosion in time increases to about 1% annually until age 60. From there it jumps to over 1% a year.

Comparing the “average” male’s declining times versus the declines seen in the World Record (which form the basis of the age-grading curve) is a bit of a two-step process. Bernard Legat holds the the WRs for M35 (3:32.51 at age 36) and M40 (3:41.87 at age 40), followed by the UK’s Anthony Whiteman (M45), and David Heath (M50) and Australia’s Keith Bateman (M55) and the US’s Nolan Shaheed (M60).

It’s interesting to track the “gap” between Mr. Average and the WR. All through the M30s, M40s, and M50s, the “gap” (as measured by the percent the average is slower than the WR) is a pretty consistent 33%. From M60 on the average man slows down at an increasing rate vs the Legats of the world so that by age 85 we are 63% slower. You can see this in the chart below.

Click    Mens1500m                          to blow it up to full size and look at the pinkish bars on the bottom of the chart. They track in percentage terms how much slower the average is than the WR.

But let’s dig a bit deeper. Let’s look at the 90th percentile*. These are the top 10% of 1500m runners in the world in their age group. The 90th percentile for M55 1500 is about 4:37.65. How well does the 90th percentile of 1500m runners (essentially those who could possibly medal at an Outdoor World Championship … see here how I arrived at that conclusion) fare? Well blow up that chart again.    Mens1500m

Below the pink bars tracking Mr. Average’s greater times than the WR you will see a line called “90th PC Diff (%)” … the percentage slower than the WR of the 90th percentile runner across the age-groups. Note how consistent it is right up to M85 … 11%-14% slower right through time. The top 10% runners really are different! Right through until their mid 80s they hold their relative performance versus the absolute best in the world.

So if the 90th percentile pretty well tracks the WR decline in performance, where does the “royal jelly” in endurance begin to slip? On the chart, just below the 90th percentile you can find the 75th percentile, which is about 5:25.59 for M60. The 75th percentile is about what it takes to qualify for a World Championship final in the 1500. For the 75th percentile, the difference vs the world record also tracks pretty darn consistently until M60. From there the percentage behind the WR increases by 2 points or so every age group. The royal jelly is seeping out. It is somewhere around here at M60 that tracking the WR begins to no longer truly reflect “everyman’s” changing performance in an endurance event through time.

Where does that leave us? I suggest you find out exactly where YOU are. What percentile is your 1500m? To find out just just click below, select your age-group from the drop down menu, select event (e.g. 1500m), and then enter your time in minutes, seconds, and hundredths of seconds. Then click the green “Ok … Done … Go” and see your standing.

Track DE

Once you know your percentile, peruse the following table. It gives the average ANNUAL decline in performance over time for the 90th percentile, the 75th, and the 50th (the average man). Your standing is probably close to one of those, so you should be able to roughly establish what kind of decline in your 1500m time you can expect over the next few years. (One proviso. I suspect that you will decline less in the first year or two of an age group, and then probably a greater percentage loss as you get to the latter years of the age-group.) After age 60 you probably should not be upset if your age grade no longer is holding with previous years. You likely are maintaining your standing among your non-elite peers.

Men's 1500m Trends and Average Annual Decline in Performance

Age-GroupWorld RecordAvg WR Decline in Performance/YearAvg 90th PC Decline in Performance/YearAvg 75th PC Decline in Performance/YearAvg Male 1500 (50th PC) Decline in Performance/Year
M353:32.51
n/an/an/an/a
M403:41.87
0.88%
1.18%
0.81%
0.47%
M453:50.55
0.78%
0.72%
0.63%
0.54%
M503:58.26
0.67%
0.60%
0.80%
0.99%
M554:12.35
1.18%
0.63%
0.88%
1.12%
M604:24.00
0.92%
1.35%
1.32%
1.30%
M654:39.87
1.20%
1.16%
1.50%
1.80%
M704:52.95
0.93%
1.31%
1.50%
1.67%
M755:22.40
2.01%
1.46%
1.82%
2.13%
M805:47.35
1.55%
1.53%
2.35%
3.11%
M856:27.30
2.30%
4.11%
3.74%
3.40%

Good luck!

Next up more information on Women’s 1500m.

To be notified immediately for the next post, click on the “follow” button at the bottom right of your screen OR become a duncanSCORE friend on Facebook here 

  • * It is important to realize that when I refer to “90th and 75th percentiles” these are statistical reference points and do not necessarily refer to actual performances or averages of actual performances.

The Aging Athlete (5) ENDURANCE – 1500M

Photo courtesy of Rob Jerome

It’s time now to see what aging does to our ability to continue on at an above “normal” pace … our endurance. And the 1500m (the “metric mile”) is as good a place as any to start that analysis.

While the 1500m certainly packs a speed element into it, by far the bigger slice of success revolves around the endurance component. Look how John Walker (never the quickest thoroughbred on the course) uses his superb endurance to win the 1500m in the Montreal Olympics.

That’s all fine and good! But once we have endurance, how long do we hold on to it? And to what extent? How fast does it slip away?

First, let’s look at the trend of the 1500m as we age. This chart shows the trend of the AVERAGE “best performance” times from mastersrankings.com (2013-2016). [For new readers to this blog and the background behind this site, you can go here to read and understand how the statistics I use are pulled together.] The chart compares Men’s and Women’s 1500m averages over time. Get a blown up look at the chart by clicking here.

MW 1500m

All times are in seconds.

You will probably note, both the Men’s and Women’s curves are quite flat in the beginning. For Men, the average ANNUAL decline is somewhere around 1/2% (see the table below)  until M45 .(Note the “annual” decline I refer to is simply the arithmetic average of the loss of performance from the previous age-group divided by 5. It’s not foolish to assume the majority of that decline is in the final 2 years of the age-group, while the first 2-3 years’ performance hit is likely to be much less pronounced.)  At M50 the yearly decline doubles to 1%+ up to M60, then edges upward until it reaches over 2% at M75, then over 3% annually thereafter.

.For Women 1500m runners, the average ANNUAL slowdown begins very much like the Men. Performance slips 1/2%-3/4% every year until it hits 1% at W55. Then the decline reaches 1.8% (W60), eases back smartly at W65 (1.25+%), and then starts to increase again thereafter You can see all the details in the table below..

Age-GroupMen's Avg 1500mM Avg Decline per YearWomen's Avg 1500mW Avg Decline per Year
M-W 354:48.68
n/a5:40.28
n/a
M-W 404:55.40
0.47%
5:46.87
0.39%
M-W 455:03.32
0.54%
5:58.94
0.70%
M-W 505:18.34
0.99%
6:12.08
0.73%
M-W 555:36.20
1.12%
6:31.57
1.05%
M-W 605:58.07
1.30%
7:07.06
1.81%
M-W 656:30.39
1.80%
7:34.08
1.27%
M-W 707:03.01
1.67%
8:28.48
2.40%
M-W 757:47.97
2.13%
9:44.77
3.00%
M-W 809:00.69
3.11%
10:42.29
1.97%
M 8510:32.50
3.40%
n/an/a

To sum up on our ability to hold our endurance … we lose 1/2%-1% per year until we hit age 55. Then the next 10 years (Men to M70) our performance will decline somewhere between 1.3% and 1.8% annually. In our 70s the performance decline gets a little faster … 2%-3% per year.

You know what? That’s really not too bad at all!

Next we will go into more detail on the Men’s 1500m

And you can, of course, easily see how your 1500m rates versus your age-group peers by checking out your duncanSCORE. Simply click here , enter your age-group and time, and you will see how you score against all in your age-group, not just against the elite of the elite.

 

To be notified immediately for the next post, click on the “follow” button at the bottom right of your screen OR become a duncanSCORE friend on Facebook here 

 

The Aging Athlete (4) – Men’s 100m

Photo courtesy Dan Slovitt

The previous posting analyzed the available statistics on Masters Women’s 100m and tried to make some sense of all the numbers. Let’s do the same with the Men’s 100m and follow where the numbers lead.

We have charted the average time for Men’s 100m by age group (in blue) versus the existing World Record (in red) for each age-group (note the M90 average is blank because the number of performances is fairly low from our reporting years 2013-2016). See full size Mens100m

One thing to note is the early Men’s WRs (M35 and M40) which are 9.92 and 9.93, held by 2 incredible Olympians (Justin Gatlin and Kim Collins). Secondly, except for the big increase between M40-M45 in the WR (and that is due to the incredibly small increase of 0.01 second in the WR for M40 by Kim Collins), the upward slope (ie the decline in performance) is gentler for the WR line than it is for the “average” male sprinter. But both lines slope rather gently (certainly more gently than Women’s) until M80 in the average line, and M90 for the WR.

As the table below shows, the average ANNUAL decline in 100m performance is very similar if you look at the WR and at the average as I have calculated from mastersrankings.com. Close, but generally we slow down a little bit more for the “average” than the WR might indicate. That is until M75 when the difference becomes much more pronounced (and perhaps this is partially due to the decreasing participation rate in the older age-groups)

Typically your performance will erode around 1/2% per year until M50, 3/4%-1% or so annually until M70, and then  1 1/2-2% a year until you hit 85.

Age-GroupWorld RecordWR Avg Annual DeclineMen's Avg 100m TimeAvg Annual Decline
M359.92n/a
12.58n/a
M409.930.02%
12.930.55%
M4510.721.59%
13.180.40%
M5010.880.30%13.590.61%
M5511.300.77%
14.130.80%
M6011.700.71%
14.820.97%
M6512.311.04%
15.811.35%
M7012.770.75%
16.300.61%
M7513.491.13%
17.551.53%
M8014.351.28%
19.442.16%
M8515.081.02%
21.552.17%
M9017.533.25%
n/an/a

So as I tried to point out in the Women’s case, even though your training is good and consistent, as you move into a different age-group, your age-grading may slowly get worse, and you may not understand why.

Here’s the answer. Not to worry. You’re typical!

I plan to investigate this with more cuts at the various percentiles to see where the changes begin to occur between “elite” and average. I’m very curious about this.

But my next project is to look at the decline over time in endurance. So next up will be the 1500.

And you can, of course, easily see how you rate versus your age-group peers by checking out your duncanSCORE. Simply click here , enter your age-group and time, and you will see how you score against all in your age-group, not just against the elite of the elite.

 

To be notified immediately for the next post, click on the “follow” button at the bottom right of your screen OR become a duncanSCORE friend on Facebook here 

 

 

The Aging Athlete (3) – Women’s 100m

Photo courtesy Dan Slovitt

The previous post summarized and trended some of the 100m data (2013-2016) from mastersrankings.com I have begun analyzing. We will look in more depth at the Men’s results soon, but first let’s explore some of the finer points of all the available performances in the Women’s 100m. For a refresher on how the data is gathered and processed, please go here.

As pointed out in the previous post, 8,626 Women’s 100m “best annual” performances are included in this analysis (772 wind assisted performances not included). An issue for Masters is how to encourage greater female participation. Unfortunately, women participate in our sport at the Masters level in far fewer numbers than men. Hopefully, sprint role models such as Karla Del Grande, Carol Layfayette-Boyd, Irene Obera and other top women’s competitors can encourage more women to do the training, don a pair of spikes, and try a race or two..

This chart graphs the W100m World Record against the average performance from mastersrankings.com. I have indicated in red the averages for W80 and W85, but since there really aren’t a lot of performances in these age-groups, I have not included W80 and W85 in my calculations. (They are there more as indications rather than  absolute data points) You can see the graph in full size by clicking

W100m for blogpost

The average 100m for all women is shown in the green curve, and the women’s World Record is shown in the mauve. Both curves take a noticeable climb upwards beginning at W70.  Also note the figures below the curves. These are the 5-year average declines in performance vs the previous age-group, for world records and all-Women averages. We will discuss these average declines in more detail from the table below, but for now here are a couple of points to note 1) both in the all-Women average and the WR, the declines moderate in intensity at W65, then increase again at W70. 2) The WR seems to indicate that this moderation occurs again at W75 (though the all-Women does NOT).

The table below takes the average 5-year decline and converts that to an estimated ANNUAL decline for the WR (column #3) and all-women’s average (column #4). You often hear Masters performance declines about 1% per year. Well, not exactly. Up to and including W50, the annual decline (as shown for both the WR and all-women’s average) is usually somewhere between 4/10 and 7/10 of a per cent. But beginning at W55, the all-women’s decline rapidly increases past 1% annually (at W65 it is 0.95% so pretty close to 1%) or greater. And for W65 onward, the all-women’s average slow down is more pronounced than that indicated by the WR. It would seem the very top athletes age differently than the “norm”.

Women's 100 Metres

Age - GroupWorld RecordAvg "Best" PerformanceWR Avg Decline/YearAvg "Best" Avg Decline/Year
W3510.74
14.89N/AN/A
W4011.0915.190.65%0.41%
W4511.3415.510.45%0.42%
W5011.6716.070.58%0.72%
W5512.8017.131.94%1.32%
W6013.6318.201.30%1.26%
W6513.9119.070.41%0.95%
W7014.7320.681.18%
1.69%
W7515.0322.480.41%
1.74%
W8016.81excl (not enough data)2.37%
N/A
W8519.83excl (not enough data)3.59%
N/A

 

 

We can look at this another way. As shown in the table below, the all-women’s average 100m time was 37-38% slower than the WR up to W50. For W55 and W60, perhaps surprisingly, the differential dropped to 33%, but from then on the differential escalates rapidly to 50%.

World Record 100mAverage 100mDifference (%)
W35 10.74

14.89
38.60%
W40 11.09
15.19
37.00%
W45 11.34
15.51
36.79%
W50 11.67
16.07
37.68%
W55 12.80
17.13
33.80%
W 60 13.63
18.20
33.56%
W 65 13.91
19.07
37.07%
W 70 14.73
20.68
40.39%
W 75 15.03
22.48
49.56%

My concluding thoughts? Up to and including W50, your performance likley will decline less than 1% per year. After that expect more than 1% per year.

And as women hit the W65 age-group, unless you are an elite or near elite, you shouldn’t be terribly disappointed if your Age Grade percentage slips a little. This would seem to be likely for the “average” or near average female sprinter. So carry on!

And you can, of course, easily see how you rate versus your age-group peers by checking out your duncanSCORE. Simply click on “Track DE”, enter your age-group and time, and you will see how you score against all in your age-group, not just against the elite of the elite.

 

Track DE

To be notified immediately for the next post, click on the “follow” button at the bottom right of your screen OR become a duncanSCORE friend on Facebook here 

 

 

 

 

 

 

 

 

The Aging Athlete (2) – 100m

How do Sprinters age? When does the aging effect on our speed really begin to increase its decline? Is the timing different women vs men? It’s probably neither consistent nor smooth. I’ve often heard it said that we, on average, lose 1% a year. Is this speed? Or endurance? Or both? I don’t know, but let’s see what the data appears to say.

I’m going to start with the 100m, which should be a good reflection of how “speed” declines with age.  I am using the duncanSCORE data base which contains 8,626 Women’s 100m entries up to and including W75 and 21,540 Men’s 100m performances (from mastersrankings.com for the years 2013-2016) used to calculate our scores and percentiles (if you are interested, you can get the details of the data processing here). Beyond W75  and M85 the numbers of performances are too sparse to use in our analysis.

The above graph may be a bit small to view properly. Here it is in full size Average    

What we see are some differences Women (in green) vs Men (in blue). Both genders decline in performance (vs the previous age group) generally about the same rate until “55”. This is shown in the tabular section under the graph lines and is labeled “% to Prev” (the percentage decline vs the previous age-group).   At that stage Women slow down 6.6% vs W50s, while M55s are averagely 3.99% slower than M50s.

The Men’s slowdown continues to accelerate (reaching 6.74% slower than the previous age group at M65), but then a sort of miracle happens! See where the blue line flattens a bit? The Men’s rate of decline (3.07%) is less then half the previous, but then accelerates much faster at M75 onward.

Women get their “mini miracle” to happen at W65. There the speed decline does indeed decline (to 4.74% from 6.29%), From there as you can see in the graph, the line begins its 45 degree upward slope. The Men’s roughly 45 degree slope commences at M75.

That’s it in a nutshell. I hope I haven’t bored you to tears, because the next posting will look at the Women’s 100m in more detail.

And don’t forget, to see how you compare against other runners in your age-group, enter your time for your event and see how your performance rates

http://duncanscore.com/track-de/

To be notified immediately for the next post, click on the “follow” button at the bottom right of your screen OR become a duncanSCORE friend on Facebook here 

If you are new here and this is all a little confusing and really don’t get what I’m talking about, please have a quick read that will explain the concepts behind the duncanSCORE

Delving Into The Aging Athlete (1)

We all age. That is if we are lucky enough to live so long. And supposedly, we all age at different rates. Probably that is true, but the data showing that is surprisingly light. Tracking aging is particularly important (and interesting!) for Masters athletes.

I am hoping to be able to provide some basic information on how we age as track and field athletes. Unfortunately, what we won’t be able to quantify is the slowdown by individual year ie how much is a 49 year old slower (on average) than a 48 year old. One day I hope enough data will be available for that.

And how does it differ Men versus Women? Or maybe it doesn’t? We will definitely look into that.

And perhaps in the not too distant future, we can look at how top performers tend to do over advancing years versus the average Master. I’m very interested in that.

We will start with the 100m, men and women, using 4 years of mastersrankings.com data (2013-2016). Then we will move on to some other events.

But first 100m. Stay tuned.

To see how your 100m time compares with your age cohort peers click here

And to be notified immediately for the next post, click on the “follow” button at the bottom right of your screen OR become a duncanSCORE friend on Facebook here

Similar Thinking

Picture from One Flew Over the Cuckoo’s Nest (1975), distributed by United Artists

Sometimes it feels a bit lonely. Like being in the wilderness and shouting alone. To the best of my knowledge, creating and using a “standardized score” for Masters Athletics has not been done before. Previously, I outlined the reasons that led me on this path.

While I might think it’s pretty darn logical to wonder how you rank within your peer age group, traditional age grading has been in use for 30 years or so, and even if its creation is often misunderstood, and one of the most common ways it is used not approved, it nonetheless is ingrained in Masters athletics. It’s why I term the duncanSCORE an alternative. Not a replacement.

And that’s why it’s interesting and a breath of fresh air to find others who share much of my thinking. A young athlete (Alexis Spinetta) has put up on her blog a very good critique of age grading and done some pretty cute calculations to derive aging tables for marathoners. Check out their thinking here … http://agegradecalculator.comabout.php. 

16 years of finishers’ times for the Austin, Texas marathon have been combined and analyzed into some pretty interesting statistics. And try out your marathon finishing time in their calculator to see how you compare against your age cohort from Austin finishers, and all other finishers.

And don’t forget if you’d like automatic notification for new blog posts, just click on the follow button  at the bottom right of your screen OR if you are on Facebook become a duncanSCORE friend here

The Numbers on Charles Allie

Charles Allie winning gold in the 400m Malaga, 2018

Photo credit courtesy of Rob Jerome

I’ve done a couple of posts on Canada’s W65 superstar Karla Del Grande, who won 3 Sprint golds at Malaga. And in the same stratosphere I would be remiss if we didn’t talk about the U.S’s unequalled M70 Sprint king Charles (Onespeed) Allie. Charlie, too, ran off with all the gold hardware in the 100-200-400 trifecta in Malaga with incredible 12.81 (in the semi) 100, 26.29 (also in the semi) 200, and a world record 57.26 (in the prelim!) in the 400.

That 12.81 100m  was only off equaling the current WR (Bobby Whilden 2005) by 4/100 of a second (note the average blink of an eye is 3-4 TENTHS of a second)..

As of this writing (December, 2018) Charles holds Outdoor World records in the 200m (M65, 24.65; M70, 25.75), 400m (M55, 52.24; M65, 56.09; M70, 57.26) and Indoor World in the 200m (M70, 26.45), 400m (M55 [coholder], 53.20; M70, 59.43). A former WMA Best Masters Athlete of the year (2013), Charlie was USATF Masters Male Athlete of the year this year (2018). He’s had quite a year!

So how do Charlie’s best times look in terms of his duncanSCORES?. His M70 times are incredibly high … 200m yields a 974score /97 percentile … 400m 978 score /98 percentile … and a 12.81 in the 100m in Malaga gives a 989 score and 99 percentile!

Beyond the analysis of the numbers, Charles’ 400m time (57.26) at age 71 is quite simply stunning. When (for a male) running your age (in years) in the 400 (in seconds) is a remarkable feat (only 23 M70s across the globe have accomplished it this year according to mastersrankings.com), running 14 seconds below your age is mind blowing!

Congratulations Charlie on a phenomenal year and great health and running in 2019!

You can find out how you rate versus your age group peers in your track event by clicking on the this link (sorry Mile and 3000m not yet available)

Track DE

UPDATE: Charlie  has been nominated for WMA Athlete of the year (and Karla Del Grande as Female Sprinter of the year) by NCCWMA (North America/Caribbean) . You can find the complete list of nominations for WMA AOTY and Categories here

And if you are unfamiliar with the concept behind the duncanSCORE and how it’s derived, go here

And don’t forget, you can get updates on the blog posts automatically by just clicking on the “follow” button on the bottom of your screen.

Karla Kontinued

Karla W65 400 Final Malaga Rob JeromePhoto courtesy of Rob Jerome

Now, I think, is an appropriate time to update what’s been happening at the World Champs in Malaga for Canada’s W65 Sprinter Karla Del Grande. I wrote a few weeks ago about her newly established record times, and compared the results duncanSCORE vs Age Grading. You can refresh your memory here.

Karla set a hot Malaga on fire. If you check the results, they list her gold medal time of 14.04 in the 100m, and her 28.83 gold medal winning time as world records. These performances are actually slightly slower to times posted earlier this season, and the applications for these times to be world records are pending. Karla ran a 100m in 13.91 at the NACAC (North/Central America/Caribbean) meet and a 200m in 28.53 at the Canadian Masters Championships. In winning the gold in the 400m in Malaga, she just missed equally the existing WR. Karla ran 68.22.

For the record, this Outdoor season Karla set 2 W65 World Records (pending) in the 100m (13.91) and 200m (28.53), and just missed (0.01 seconds,1:06.22) equaling the World Record in the 400m.

Also for the record, those times have Age Grades of 96.84% (100m), 98.13% (200m), and 96.89% (400m). AG rates her 200m as the best, and 100m and 400 near equal.

The duncanSCORE evaluates these performances slightly differently. 967/97 percentile (100m), identical 967/97 percentile for the 200m, and 952/95 percentile for the 400m. That’s an equal performance in the 100m and 200m and and a 400m about 2% less.

As well, these new W65 world records are rated by AG as inferior to her W60 world records from 5 years ago. Karla’s duncanSCORES on the other hand rate them significantly superior.

I’m biased I admit. But I think the dSCORES are a better evaluation.

And don’t forget, you can get updates on the blog posts automatically by just clicking on the “follow” button on the bottom of your screen.

Distance (Track), Hurdles, & Steeplechase

Note: photo sourced from http://potchefstroomherald.co.za, Photographer unknown

 

It’s time to finish out this series with the track events away from the Sprints and Middle Distance.

Specifically the Sprint Hurdles, Long Hurdles, Steeplechase, and 5000m and 10000m.

As before for the Sprints, and Middle Distance,  I am going to try and estimate what duncanSCORE percentile you likely need to reach the podium at a WMA World Championship. And like previously, I am using the results from the last 4 WMA Outdoor Championships (Sacramento 2011, Porto Alegre 2013, Lyon 2015, and Perth 2016). Results have been tabulated from M/W 35 up to and including M/W 75. (If you are unfamiliar with the duncanSCORE as an alternative to age-grading, you can get a quick briefing here.

There is certainly a lot more variation in these events than in the mainstream Sprints and Middle Distance. As well, getting to the final is automatic in the Steeple and 5000 and 10000, and a much lower duncanSCORE percentile is needed to get to the final in Sprint Hurdles (69%) and the Long Hurdles (65%). ,

Ah, but climbing on the podium you ask? What does that take? You might remember that to win at least a Bronze medal in the Sprints, you probably need to be able to run the distance at a 92 percentile. 800 or 1500 usually means 90-91 percentile. And remember too, that the Steeple and  5000m and 10000m are single off events … no prelims or semis … so weather can be a big factor. What I’m saying is there is more variation in the Hurdles and Steeple. Estimating the requirements for a medal is not as consistent as the other running events. But based on the last 4 WC, to medal in the Sprint Hurdles, you will need an 85 percentile. Long Hurdles 82%. And all you Steeplers out there … 81%!

Lots of numbers to remember … so here is a table summarizing all these key dSCORES by event. Just click here! (in MS Excel)

dSCORE Summary by Event

And to get your personal duncanSCORE and percentile for these events, go here for the calculation