# Example 852 — The ages were bucketed and the mean used the midpoints `the_ages_were_bucketed_and_the_mean_used_the_midpoints.eml` - A survey reports the average age, computed as the weighted mean of the age-bucket midpoints, and the arithmetic is correct over every respondent. What a bucket midpoint stands in for is computed below. ## EML ```eml # Self-authored for the EML case corpus (no external origin). A survey reports the # average age, computed as the weighted mean of the age-bucket midpoints, and the # arithmetic is correct over every respondent. What a bucket midpoint stands in # for is computed below. # # The measurement is careful. It reads the real recorded bucket for each # respondent, not a guess; it covers every respondent; the weighted mean of the # midpoints is honest; and the intent is exactly 'the average age'. # # Only the bucket is recorded, never the age, so each respondent is placed at the # center of the bucket - which is the true average only if ages sit symmetrically # in every bucket, and the top bucket is open-ended with a chosen midpoint. 100000 => respondents 42 => mean_from_bucket_midpoints 47 => mean_from_raw_ages mean_from_raw_ages - mean_from_bucket_midpoints => years_the_bucketing_hid int(years_the_bucketing_hid * 10000 / mean_from_raw_ages) => understated_share_per_myriad "respondents : " + str(respondents) ^0 "mean from bucket midpoints : " + str(mean_from_bucket_midpoints) ^0 "mean from the raw ages : " + str(mean_from_raw_ages) ^0 "years the bucketing hid : " + str(years_the_bucketing_hid) ^0 "understated share of the truth : " + str(understated_share_per_myriad) + " per ten thousand" ^0 "" ^0 # ---- what the measurement verified ---- "the average-age computation" ^0 " reads : the real recorded bucket for each respondent" ^0 " covers : every respondent" ^0 " mean : the honest weighted mean of the midpoints" ^0 " intent : the average age" ^0 " respondents omitted : 0" ^0 " verdict : MIDPOINT MEAN IS 42, COMPUTED CORRECTLY" ^0 "" ^0 " taking the honest weighted mean of the midpoints over" ^0 " every respondent is the part done right here, and it is" ^0 " why 42 is the correct mean of the midpoints" ^0 "" ^0 # ---- what a midpoint stands in for ---- "the midpoint substitution" ^0 " what is recorded : the bucket, never the age" ^0 " what the midpoint assumes : the age sits at the bucket's" ^0 " center" ^0 " when that holds : only if ages are symmetric within each" ^0 " bucket" ^0 " the real distribution : skews old, and the top bucket is" ^0 " open-ended with a guessed center" ^0 " so the midpoint mean : is the mean of the assumption, not" ^0 " of the ages" ^0 "" ^0 # ---- what the caller got ---- "the result of the survey" ^0 " reported average age : " + str(mean_from_bucket_midpoints) ^0 " average from the raw ages : " + str(mean_from_raw_ages) ^0 " years hidden by the bucketing : " + str(years_the_bucketing_hid) ^0 " is the midpoint mean miscomputed : no; 42 is exact for" ^0 " the midpoints" ^0 " is 42 the average age : no; interval censoring replaced" ^0 " each age with a center it need not sit at" ^0 "" ^0 # ---- null control ---- # The same respondents, with the raw age recorded instead of only the bucket (or a # within-bucket distribution estimated rather than assumed uniform). 42 => nc_mean_from_midpoints 47 => nc_mean_from_raw_ages 5 => nc_years_the_raw_value_recovers "null control - record the raw age, not only the bucket" ^0 " mean from midpoints : " + str(nc_mean_from_midpoints) ^0 " mean from raw ages : " + str(nc_mean_from_raw_ages) ^0 " years the raw value recovers : " + str(nc_years_the_raw_value_recovers) ^0 " no respondent and no bucket boundary changed; each age" ^0 " stopped being read as its bucket's center and started" ^0 " being read as itself" ^0 "" ^0 # ---- the rule ---- "what a mean of bucket midpoints guarantees" ^0 " it is the correct mean of the midpoints : exactly, real" ^0 " buckets, every respondent, honest weighting" ^0 " it is the mean of the ages : not addressed; only the" ^0 " bucket is recorded, so each age becomes its bucket's" ^0 " center, and a skewed distribution plus an open top" ^0 " bucket moves the true mean to " + str(mean_from_raw_ages) ^0 "" ^0 "a bucket keeps which interval a value fell in and forgets where inside it; the" ^0 "midpoint puts it back at the center, which is a fact about the interval, not the" ^0 "value, and a skew within the bucket is invisible to a mean of centers" ^0 "" ^0 "It takes the honest weighted mean of the midpoints over every respondent - 42 is" ^0 "the exact midpoint mean. But only the bucket was recorded, so each age is read as" ^0 "its center; with a skewed distribution and an open top bucket the true mean is " + str(mean_from_raw_ages) + "," ^0 "" + str(understated_share_per_myriad) + " per ten thousand understated, until the raw age is recorded." ^0 ``` ## Python (deterministic transpilation) ```python respondents = 100000 mean_from_bucket_midpoints = 42 mean_from_raw_ages = 47 years_the_bucketing_hid = mean_from_raw_ages - mean_from_bucket_midpoints understated_share_per_myriad = int(years_the_bucketing_hid * 10000 / mean_from_raw_ages) print("respondents : " + str(respondents)) print("mean from bucket midpoints : " + str(mean_from_bucket_midpoints)) print("mean from the raw ages : " + str(mean_from_raw_ages)) print("years the bucketing hid : " + str(years_the_bucketing_hid)) print("understated share of the truth : " + str(understated_share_per_myriad) + " per ten thousand") print("") print("the average-age computation") print(" reads : the real recorded bucket for each respondent") print(" covers : every respondent") print(" mean : the honest weighted mean of the midpoints") print(" intent : the average age") print(" respondents omitted : 0") print(" verdict : MIDPOINT MEAN IS 42, COMPUTED CORRECTLY") print("") print(" taking the honest weighted mean of the midpoints over") print(" every respondent is the part done right here, and it is") print(" why 42 is the correct mean of the midpoints") print("") print("the midpoint substitution") print(" what is recorded : the bucket, never the age") print(" what the midpoint assumes : the age sits at the bucket's") print(" center") print(" when that holds : only if ages are symmetric within each") print(" bucket") print(" the real distribution : skews old, and the top bucket is") print(" open-ended with a guessed center") print(" so the midpoint mean : is the mean of the assumption, not") print(" of the ages") print("") print("the result of the survey") print(" reported average age : " + str(mean_from_bucket_midpoints)) print(" average from the raw ages : " + str(mean_from_raw_ages)) print(" years hidden by the bucketing : " + str(years_the_bucketing_hid)) print(" is the midpoint mean miscomputed : no; 42 is exact for") print(" the midpoints") print(" is 42 the average age : no; interval censoring replaced") print(" each age with a center it need not sit at") print("") nc_mean_from_midpoints = 42 nc_mean_from_raw_ages = 47 nc_years_the_raw_value_recovers = 5 print("null control - record the raw age, not only the bucket") print(" mean from midpoints : " + str(nc_mean_from_midpoints)) print(" mean from raw ages : " + str(nc_mean_from_raw_ages)) print(" years the raw value recovers : " + str(nc_years_the_raw_value_recovers)) print(" no respondent and no bucket boundary changed; each age") print(" stopped being read as its bucket's center and started") print(" being read as itself") print("") print("what a mean of bucket midpoints guarantees") print(" it is the correct mean of the midpoints : exactly, real") print(" buckets, every respondent, honest weighting") print(" it is the mean of the ages : not addressed; only the") print(" bucket is recorded, so each age becomes its bucket's") print(" center, and a skewed distribution plus an open top") print(" bucket moves the true mean to " + str(mean_from_raw_ages)) print("") print("a bucket keeps which interval a value fell in and forgets where inside it; the") print("midpoint puts it back at the center, which is a fact about the interval, not the") print("value, and a skew within the bucket is invisible to a mean of centers") print("") print("It takes the honest weighted mean of the midpoints over every respondent - 42 is") print("the exact midpoint mean. But only the bucket was recorded, so each age is read as") print("its center; with a skewed distribution and an open top bucket the true mean is " + str(mean_from_raw_ages) + ",") print("" + str(understated_share_per_myriad) + " per ten thousand understated, until the raw age is recorded.") ``` ## stdout (executed) ```text respondents : 100000 mean from bucket midpoints : 42 mean from the raw ages : 47 years the bucketing hid : 5 understated share of the truth : 1063 per ten thousand the average-age computation reads : the real recorded bucket for each respondent covers : every respondent mean : the honest weighted mean of the midpoints intent : the average age respondents omitted : 0 verdict : MIDPOINT MEAN IS 42, COMPUTED CORRECTLY taking the honest weighted mean of the midpoints over every respondent is the part done right here, and it is why 42 is the correct mean of the midpoints the midpoint substitution what is recorded : the bucket, never the age what the midpoint assumes : the age sits at the bucket's center when that holds : only if ages are symmetric within each bucket the real distribution : skews old, and the top bucket is open-ended with a guessed center so the midpoint mean : is the mean of the assumption, not of the ages the result of the survey reported average age : 42 average from the raw ages : 47 years hidden by the bucketing : 5 is the midpoint mean miscomputed : no; 42 is exact for the midpoints is 42 the average age : no; interval censoring replaced each age with a center it need not sit at null control - record the raw age, not only the bucket mean from midpoints : 42 mean from raw ages : 47 years the raw value recovers : 5 no respondent and no bucket boundary changed; each age stopped being read as its bucket's center and started being read as itself what a mean of bucket midpoints guarantees it is the correct mean of the midpoints : exactly, real buckets, every respondent, honest weighting it is the mean of the ages : not addressed; only the bucket is recorded, so each age becomes its bucket's center, and a skewed distribution plus an open top bucket moves the true mean to 47 a bucket keeps which interval a value fell in and forgets where inside it; the midpoint puts it back at the center, which is a fact about the interval, not the value, and a skew within the bucket is invisible to a mean of centers It takes the honest weighted mean of the midpoints over every respondent - 42 is the exact midpoint mean. But only the bucket was recorded, so each age is read as its center; with a skewed distribution and an open top bucket the true mean is 47, 1063 per ten thousand understated, until the raw age is recorded. ``` ## Round-trip `ok: true` — round-trip fixpoint reached (python1 == python2) ## Trace event types eml:run:start · eml:assign · eml:output · eml:run:done