Showing posts with label Standard Deviation. Show all posts
Showing posts with label Standard Deviation. Show all posts
Jun 24, 2011
Men's Body Weight Distribution
On my quest for determining how much people can lift, I developed this graph of men's body weights in pounds. The natural log of men's weights in kilograms follows a normal distribution with a standard deviation of 0.165. The equation I used, based on the EPA's data on the weights of American men ages 18-74, is e^(Standard_Deviations * 0.165 + 4.34) in kilograms. The proportion of men of each weight is in percent.
Some RPG systems have rules regarding character weight. For example, Heavy Gear and Shadowrun 4th ed. have attributes for Size or Body, respectively, based on the size of a person's body. In real life, for the purposes of determining strength, the maximum amount of weight that a man can lift is close to 3.5 times his weight. D&D is ridiculous for allowing halflings to have a STR of 18 at creation.
Jun 23, 2011
Strength and Untrained Lifting in Aberrant
I am crunching data on human strength for a series of posts on how TTRPGs model it. This post can be written early as a teaser because White Wolf's Aberrant rules separate out lifting capacity by untrained natural ability and the product of training. Untrained strength in Aberrant has a scale from 1 to 5, with 2 being average. There is no real standard deviation for the Aberrant stats, but it seemed reasonable enough to use each ability level as a proxy.
As I will go into in more detail in a later post, the best resource I found for lifting ability shows average weight lifted for each body weight category. While trying to find distributions of adult male body weight so that I could determine the proportion of men who could lift each amount of weight, I found a couple differing sets of numbers from the EPA and WolframAlpha, and a set for older men from an organization in Cambridge, MA. The problem with weight is that it does not follow a normal distribution, so I cannot use the mean and standard deviation to calculate any value. However, the natural log (ln) of weight does, and that may help me. The differences between the data sets seem small enough not to significantly affect my work, and are at least partially explained by the general rise in Americans' weights over the last couple decades. For the graph above, the x-axis is technically mislabeled, but each standard deviation mark refers to the percentile for that z-score, so -1 SD is about the 16th percentile, 1 SD is the 84th, 2 SD is the 97.7th, and 3 SD is about the 99.9th.
As you can see, the Aberrant rules are pretty bad at simulating real untrained strength among men. When I finish figuring out what the distribution of strength is for all men, women, and adults together, I will show how each RPG system performs as a simulator, and I will revisit the Aberrant system to include the weightlifting skill.
As I will go into in more detail in a later post, the best resource I found for lifting ability shows average weight lifted for each body weight category. While trying to find distributions of adult male body weight so that I could determine the proportion of men who could lift each amount of weight, I found a couple differing sets of numbers from the EPA and WolframAlpha, and a set for older men from an organization in Cambridge, MA. The problem with weight is that it does not follow a normal distribution, so I cannot use the mean and standard deviation to calculate any value. However, the natural log (ln) of weight does, and that may help me. The differences between the data sets seem small enough not to significantly affect my work, and are at least partially explained by the general rise in Americans' weights over the last couple decades. For the graph above, the x-axis is technically mislabeled, but each standard deviation mark refers to the percentile for that z-score, so -1 SD is about the 16th percentile, 1 SD is the 84th, 2 SD is the 97.7th, and 3 SD is about the 99.9th.
As you can see, the Aberrant rules are pretty bad at simulating real untrained strength among men. When I finish figuring out what the distribution of strength is for all men, women, and adults together, I will show how each RPG system performs as a simulator, and I will revisit the Aberrant system to include the weightlifting skill.
Jan 29, 2011
D&D and IQ
I first started playing RPGs about two decades ago with Advanced Dungeons and Dragons. Back before 4th edition finally defaulted to a point-buy system for attributes, character attributes were determined by rolling six-sided dice (d6). The basic and most scary method was to roll 3d6 and add the results, providing a total from 3 to 18. The "heroic" method was to roll 4d6 and sum the highest three results, which still provided a total from 3 to 18, but with a higher mean.
It had generally been said in my gaming circles that Intelligence (INT) in D&D was the equivalent of an intelligence quotient (IQ) divided by 10. Or INT * 10 = IQ. The average IQ is 100, and the D&D manuals stated that the average INT was 10 (really 10.5 by rolling 3d6). Heroes, rolling 4d6 and dropping the lowest, have a mode of 13 and a mean of 12.2446. The whole INT distribution, however, does not fit as well as its mean onto the real IQ distribution.
The old Stanford Binet calculated IQ as 100 times the ratio between a child's age and the age of children with the same performance on average on particular tasks. So, a 6-year old who performed as well as the average 9-year old would have an IQ of 150. This method of determining IQ scores becomes less useful as children grow up, and makes no sense to apply to adults. The newer Wechsler tests calculate IQ by taking the scores of everyone in an age group and putting them on a normal distribution with a mean of 100 and a standard deviation (SD) of 15. This is a much more clear way to evaluate how a person functions intellectually compared to same-age peers.
If we take the distribution of 3d6 rolls and associate the probability of each sum with the prevalence of corresponding IQs in the real world, we can get a better impression of what a D&D INT score means in terms of IQ. If we say that 3d6 simulates a normal distribution of intelligence, then here are the translations (rounded) of each INT score into an IQ score:
INT IQ
3 57
4 66
5 72
6 78
7 83
8 88
9 93
10 98
11 102
12 107
13 112
14 117
15 122
16 128
17 134
18 143
The IQ scores in this table are calculated from the z-scores for the middles of each range of probability that maps onto each INT score. For example, an INT of 3 and of 18 each have a 0.463 chance of occurring (1 in 216, which is 6^3), but I cannot use the z-score for .00463 to calculate IQ for an INT of 3 and work my way up because I would be left trying to use the z-score for 1 with the INT of 18, which is infinity standard deviations. Also, I would end up using the z-score for .5 for an INT of 10 when it truly applies to an INT of 10.5. What works is to use the z-scores for the midpoints between probabilities. So, an IQ of 57 (2.84 standard deviations below the mean) corresponds to the z-score for .0023 (the midpoint between 1/216 and 0, since it is impossible to roll less than 3) with some rounding. The probability of rolling at least a 9 is .375, and the probability of rolling at least a 10 is .5, the midpoint is .4375 for an INT of 10, which has a z-score of
about -.157. 100 (the mean) minus (.157 * 15 (the standard deviation)) equals about 97.645, so the IQ for an INT of 10 is 98 after rounding. Each IQ score was calculated this way.
As you can see, multiplying INT by 10 does not give you your character's IQ. Now, this does not address differences in intellectual ability by age group. A stereotypical D&D party consists of characters emerging from adolescence into adulthood, roughly in the same stage of development, so it should be okay to use this scale. A 12-year old with a 120 IQ still lacks much of the functioning of a 20-year old with a 100 IQ, and a 30-year old with a 100 IQ functions a little differently than a 50-year old with a 100 IQ. Age tends to result in a larger vocabulary and body of knowledge with a slower information processing speed, but I may go deeper into how RPGs simulate the effects of aging on intelligence in later posts.
For that matter, using an overall IQ score obfuscates the differences in each person's separate areas of intellectual functioning. "Intelligence" refers to many different abilities that we have, and a dozen people with the same IQ can each function differently from each other on a variety of tasks. Few simulation systems break down intelligence into components, and that is also a topic I will return to in future posts.
It had generally been said in my gaming circles that Intelligence (INT) in D&D was the equivalent of an intelligence quotient (IQ) divided by 10. Or INT * 10 = IQ. The average IQ is 100, and the D&D manuals stated that the average INT was 10 (really 10.5 by rolling 3d6). Heroes, rolling 4d6 and dropping the lowest, have a mode of 13 and a mean of 12.2446. The whole INT distribution, however, does not fit as well as its mean onto the real IQ distribution.
The old Stanford Binet calculated IQ as 100 times the ratio between a child's age and the age of children with the same performance on average on particular tasks. So, a 6-year old who performed as well as the average 9-year old would have an IQ of 150. This method of determining IQ scores becomes less useful as children grow up, and makes no sense to apply to adults. The newer Wechsler tests calculate IQ by taking the scores of everyone in an age group and putting them on a normal distribution with a mean of 100 and a standard deviation (SD) of 15. This is a much more clear way to evaluate how a person functions intellectually compared to same-age peers.
If we take the distribution of 3d6 rolls and associate the probability of each sum with the prevalence of corresponding IQs in the real world, we can get a better impression of what a D&D INT score means in terms of IQ. If we say that 3d6 simulates a normal distribution of intelligence, then here are the translations (rounded) of each INT score into an IQ score:
INT IQ
3 57
4 66
5 72
6 78
7 83
8 88
9 93
10 98
11 102
12 107
13 112
14 117
15 122
16 128
17 134
18 143
The IQ scores in this table are calculated from the z-scores for the middles of each range of probability that maps onto each INT score. For example, an INT of 3 and of 18 each have a 0.463 chance of occurring (1 in 216, which is 6^3), but I cannot use the z-score for .00463 to calculate IQ for an INT of 3 and work my way up because I would be left trying to use the z-score for 1 with the INT of 18, which is infinity standard deviations. Also, I would end up using the z-score for .5 for an INT of 10 when it truly applies to an INT of 10.5. What works is to use the z-scores for the midpoints between probabilities. So, an IQ of 57 (2.84 standard deviations below the mean) corresponds to the z-score for .0023 (the midpoint between 1/216 and 0, since it is impossible to roll less than 3) with some rounding. The probability of rolling at least a 9 is .375, and the probability of rolling at least a 10 is .5, the midpoint is .4375 for an INT of 10, which has a z-score of
about -.157. 100 (the mean) minus (.157 * 15 (the standard deviation)) equals about 97.645, so the IQ for an INT of 10 is 98 after rounding. Each IQ score was calculated this way.
As you can see, multiplying INT by 10 does not give you your character's IQ. Now, this does not address differences in intellectual ability by age group. A stereotypical D&D party consists of characters emerging from adolescence into adulthood, roughly in the same stage of development, so it should be okay to use this scale. A 12-year old with a 120 IQ still lacks much of the functioning of a 20-year old with a 100 IQ, and a 30-year old with a 100 IQ functions a little differently than a 50-year old with a 100 IQ. Age tends to result in a larger vocabulary and body of knowledge with a slower information processing speed, but I may go deeper into how RPGs simulate the effects of aging on intelligence in later posts.
For that matter, using an overall IQ score obfuscates the differences in each person's separate areas of intellectual functioning. "Intelligence" refers to many different abilities that we have, and a dozen people with the same IQ can each function differently from each other on a variety of tasks. Few simulation systems break down intelligence into components, and that is also a topic I will return to in future posts.
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