Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 1 | ## Module statistics.py |
| 2 | ## |
| 3 | ## Copyright (c) 2013 Steven D'Aprano <steve+python@pearwood.info>. |
| 4 | ## |
| 5 | ## Licensed under the Apache License, Version 2.0 (the "License"); |
| 6 | ## you may not use this file except in compliance with the License. |
| 7 | ## You may obtain a copy of the License at |
| 8 | ## |
| 9 | ## http://www.apache.org/licenses/LICENSE-2.0 |
| 10 | ## |
| 11 | ## Unless required by applicable law or agreed to in writing, software |
| 12 | ## distributed under the License is distributed on an "AS IS" BASIS, |
| 13 | ## WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| 14 | ## See the License for the specific language governing permissions and |
| 15 | ## limitations under the License. |
| 16 | |
| 17 | |
| 18 | """ |
| 19 | Basic statistics module. |
| 20 | |
| 21 | This module provides functions for calculating statistics of data, including |
| 22 | averages, variance, and standard deviation. |
| 23 | |
| 24 | Calculating averages |
| 25 | -------------------- |
| 26 | |
| 27 | ================== ============================================= |
| 28 | Function Description |
| 29 | ================== ============================================= |
| 30 | mean Arithmetic mean (average) of data. |
Steven D'Aprano | a474afd | 2016-08-09 12:49:01 +1000 | [diff] [blame^] | 31 | harmonic_mean Harmonic mean of data. |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 32 | median Median (middle value) of data. |
| 33 | median_low Low median of data. |
| 34 | median_high High median of data. |
| 35 | median_grouped Median, or 50th percentile, of grouped data. |
| 36 | mode Mode (most common value) of data. |
| 37 | ================== ============================================= |
| 38 | |
| 39 | Calculate the arithmetic mean ("the average") of data: |
| 40 | |
| 41 | >>> mean([-1.0, 2.5, 3.25, 5.75]) |
| 42 | 2.625 |
| 43 | |
| 44 | |
| 45 | Calculate the standard median of discrete data: |
| 46 | |
| 47 | >>> median([2, 3, 4, 5]) |
| 48 | 3.5 |
| 49 | |
| 50 | |
| 51 | Calculate the median, or 50th percentile, of data grouped into class intervals |
| 52 | centred on the data values provided. E.g. if your data points are rounded to |
| 53 | the nearest whole number: |
| 54 | |
| 55 | >>> median_grouped([2, 2, 3, 3, 3, 4]) #doctest: +ELLIPSIS |
| 56 | 2.8333333333... |
| 57 | |
| 58 | This should be interpreted in this way: you have two data points in the class |
| 59 | interval 1.5-2.5, three data points in the class interval 2.5-3.5, and one in |
| 60 | the class interval 3.5-4.5. The median of these data points is 2.8333... |
| 61 | |
| 62 | |
| 63 | Calculating variability or spread |
| 64 | --------------------------------- |
| 65 | |
| 66 | ================== ============================================= |
| 67 | Function Description |
| 68 | ================== ============================================= |
| 69 | pvariance Population variance of data. |
| 70 | variance Sample variance of data. |
| 71 | pstdev Population standard deviation of data. |
| 72 | stdev Sample standard deviation of data. |
| 73 | ================== ============================================= |
| 74 | |
| 75 | Calculate the standard deviation of sample data: |
| 76 | |
| 77 | >>> stdev([2.5, 3.25, 5.5, 11.25, 11.75]) #doctest: +ELLIPSIS |
| 78 | 4.38961843444... |
| 79 | |
| 80 | If you have previously calculated the mean, you can pass it as the optional |
| 81 | second argument to the four "spread" functions to avoid recalculating it: |
| 82 | |
| 83 | >>> data = [1, 2, 2, 4, 4, 4, 5, 6] |
| 84 | >>> mu = mean(data) |
| 85 | >>> pvariance(data, mu) |
| 86 | 2.5 |
| 87 | |
| 88 | |
| 89 | Exceptions |
| 90 | ---------- |
| 91 | |
| 92 | A single exception is defined: StatisticsError is a subclass of ValueError. |
| 93 | |
| 94 | """ |
| 95 | |
| 96 | __all__ = [ 'StatisticsError', |
| 97 | 'pstdev', 'pvariance', 'stdev', 'variance', |
| 98 | 'median', 'median_low', 'median_high', 'median_grouped', |
Steven D'Aprano | a474afd | 2016-08-09 12:49:01 +1000 | [diff] [blame^] | 99 | 'mean', 'mode', 'harmonic_mean', |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 100 | ] |
| 101 | |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 102 | import collections |
Steven D'Aprano | a474afd | 2016-08-09 12:49:01 +1000 | [diff] [blame^] | 103 | import decimal |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 104 | import math |
Steven D'Aprano | a474afd | 2016-08-09 12:49:01 +1000 | [diff] [blame^] | 105 | import numbers |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 106 | |
| 107 | from fractions import Fraction |
| 108 | from decimal import Decimal |
Steven D'Aprano | a474afd | 2016-08-09 12:49:01 +1000 | [diff] [blame^] | 109 | from itertools import groupby, chain |
Steven D'Aprano | 3b06e24 | 2016-05-05 03:54:29 +1000 | [diff] [blame] | 110 | from bisect import bisect_left, bisect_right |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 111 | |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 112 | |
| 113 | |
| 114 | # === Exceptions === |
| 115 | |
| 116 | class StatisticsError(ValueError): |
| 117 | pass |
| 118 | |
| 119 | |
| 120 | # === Private utilities === |
| 121 | |
| 122 | def _sum(data, start=0): |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 123 | """_sum(data [, start]) -> (type, sum, count) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 124 | |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 125 | Return a high-precision sum of the given numeric data as a fraction, |
| 126 | together with the type to be converted to and the count of items. |
| 127 | |
| 128 | If optional argument ``start`` is given, it is added to the total. |
| 129 | If ``data`` is empty, ``start`` (defaulting to 0) is returned. |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 130 | |
| 131 | |
| 132 | Examples |
| 133 | -------- |
| 134 | |
| 135 | >>> _sum([3, 2.25, 4.5, -0.5, 1.0], 0.75) |
Benjamin Peterson | ab078e9 | 2016-07-13 21:13:29 -0700 | [diff] [blame] | 136 | (<class 'float'>, Fraction(11, 1), 5) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 137 | |
| 138 | Some sources of round-off error will be avoided: |
| 139 | |
Steven D'Aprano | a474afd | 2016-08-09 12:49:01 +1000 | [diff] [blame^] | 140 | # Built-in sum returns zero. |
| 141 | >>> _sum([1e50, 1, -1e50] * 1000) |
Benjamin Peterson | ab078e9 | 2016-07-13 21:13:29 -0700 | [diff] [blame] | 142 | (<class 'float'>, Fraction(1000, 1), 3000) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 143 | |
| 144 | Fractions and Decimals are also supported: |
| 145 | |
| 146 | >>> from fractions import Fraction as F |
| 147 | >>> _sum([F(2, 3), F(7, 5), F(1, 4), F(5, 6)]) |
Benjamin Peterson | ab078e9 | 2016-07-13 21:13:29 -0700 | [diff] [blame] | 148 | (<class 'fractions.Fraction'>, Fraction(63, 20), 4) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 149 | |
| 150 | >>> from decimal import Decimal as D |
| 151 | >>> data = [D("0.1375"), D("0.2108"), D("0.3061"), D("0.0419")] |
| 152 | >>> _sum(data) |
Benjamin Peterson | ab078e9 | 2016-07-13 21:13:29 -0700 | [diff] [blame] | 153 | (<class 'decimal.Decimal'>, Fraction(6963, 10000), 4) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 154 | |
Nick Coghlan | 73afe2a | 2014-02-08 19:58:04 +1000 | [diff] [blame] | 155 | Mixed types are currently treated as an error, except that int is |
| 156 | allowed. |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 157 | """ |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 158 | count = 0 |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 159 | n, d = _exact_ratio(start) |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 160 | partials = {d: n} |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 161 | partials_get = partials.get |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 162 | T = _coerce(int, type(start)) |
| 163 | for typ, values in groupby(data, type): |
| 164 | T = _coerce(T, typ) # or raise TypeError |
| 165 | for n,d in map(_exact_ratio, values): |
| 166 | count += 1 |
| 167 | partials[d] = partials_get(d, 0) + n |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 168 | if None in partials: |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 169 | # The sum will be a NAN or INF. We can ignore all the finite |
| 170 | # partials, and just look at this special one. |
| 171 | total = partials[None] |
| 172 | assert not _isfinite(total) |
| 173 | else: |
| 174 | # Sum all the partial sums using builtin sum. |
| 175 | # FIXME is this faster if we sum them in order of the denominator? |
| 176 | total = sum(Fraction(n, d) for d, n in sorted(partials.items())) |
| 177 | return (T, total, count) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 178 | |
| 179 | |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 180 | def _isfinite(x): |
| 181 | try: |
| 182 | return x.is_finite() # Likely a Decimal. |
| 183 | except AttributeError: |
| 184 | return math.isfinite(x) # Coerces to float first. |
| 185 | |
| 186 | |
| 187 | def _coerce(T, S): |
| 188 | """Coerce types T and S to a common type, or raise TypeError. |
| 189 | |
| 190 | Coercion rules are currently an implementation detail. See the CoerceTest |
| 191 | test class in test_statistics for details. |
| 192 | """ |
| 193 | # See http://bugs.python.org/issue24068. |
| 194 | assert T is not bool, "initial type T is bool" |
| 195 | # If the types are the same, no need to coerce anything. Put this |
| 196 | # first, so that the usual case (no coercion needed) happens as soon |
| 197 | # as possible. |
| 198 | if T is S: return T |
| 199 | # Mixed int & other coerce to the other type. |
| 200 | if S is int or S is bool: return T |
| 201 | if T is int: return S |
| 202 | # If one is a (strict) subclass of the other, coerce to the subclass. |
| 203 | if issubclass(S, T): return S |
| 204 | if issubclass(T, S): return T |
| 205 | # Ints coerce to the other type. |
| 206 | if issubclass(T, int): return S |
| 207 | if issubclass(S, int): return T |
| 208 | # Mixed fraction & float coerces to float (or float subclass). |
| 209 | if issubclass(T, Fraction) and issubclass(S, float): |
| 210 | return S |
| 211 | if issubclass(T, float) and issubclass(S, Fraction): |
| 212 | return T |
| 213 | # Any other combination is disallowed. |
| 214 | msg = "don't know how to coerce %s and %s" |
| 215 | raise TypeError(msg % (T.__name__, S.__name__)) |
Nick Coghlan | 73afe2a | 2014-02-08 19:58:04 +1000 | [diff] [blame] | 216 | |
| 217 | |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 218 | def _exact_ratio(x): |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 219 | """Return Real number x to exact (numerator, denominator) pair. |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 220 | |
| 221 | >>> _exact_ratio(0.25) |
| 222 | (1, 4) |
| 223 | |
| 224 | x is expected to be an int, Fraction, Decimal or float. |
| 225 | """ |
| 226 | try: |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 227 | # Optimise the common case of floats. We expect that the most often |
| 228 | # used numeric type will be builtin floats, so try to make this as |
| 229 | # fast as possible. |
Steven D'Aprano | 3b06e24 | 2016-05-05 03:54:29 +1000 | [diff] [blame] | 230 | if type(x) is float or type(x) is Decimal: |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 231 | return x.as_integer_ratio() |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 232 | try: |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 233 | # x may be an int, Fraction, or Integral ABC. |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 234 | return (x.numerator, x.denominator) |
| 235 | except AttributeError: |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 236 | try: |
Steven D'Aprano | 3b06e24 | 2016-05-05 03:54:29 +1000 | [diff] [blame] | 237 | # x may be a float or Decimal subclass. |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 238 | return x.as_integer_ratio() |
| 239 | except AttributeError: |
Steven D'Aprano | 3b06e24 | 2016-05-05 03:54:29 +1000 | [diff] [blame] | 240 | # Just give up? |
| 241 | pass |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 242 | except (OverflowError, ValueError): |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 243 | # float NAN or INF. |
Steven D'Aprano | 3b06e24 | 2016-05-05 03:54:29 +1000 | [diff] [blame] | 244 | assert not _isfinite(x) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 245 | return (x, None) |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 246 | msg = "can't convert type '{}' to numerator/denominator" |
| 247 | raise TypeError(msg.format(type(x).__name__)) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 248 | |
| 249 | |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 250 | def _convert(value, T): |
| 251 | """Convert value to given numeric type T.""" |
| 252 | if type(value) is T: |
| 253 | # This covers the cases where T is Fraction, or where value is |
| 254 | # a NAN or INF (Decimal or float). |
| 255 | return value |
| 256 | if issubclass(T, int) and value.denominator != 1: |
| 257 | T = float |
| 258 | try: |
| 259 | # FIXME: what do we do if this overflows? |
| 260 | return T(value) |
| 261 | except TypeError: |
| 262 | if issubclass(T, Decimal): |
| 263 | return T(value.numerator)/T(value.denominator) |
| 264 | else: |
| 265 | raise |
| 266 | |
| 267 | |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 268 | def _counts(data): |
| 269 | # Generate a table of sorted (value, frequency) pairs. |
Nick Coghlan | bfd68bf | 2014-02-08 19:44:16 +1000 | [diff] [blame] | 270 | table = collections.Counter(iter(data)).most_common() |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 271 | if not table: |
| 272 | return table |
| 273 | # Extract the values with the highest frequency. |
| 274 | maxfreq = table[0][1] |
| 275 | for i in range(1, len(table)): |
| 276 | if table[i][1] != maxfreq: |
| 277 | table = table[:i] |
| 278 | break |
| 279 | return table |
| 280 | |
| 281 | |
Steven D'Aprano | 3b06e24 | 2016-05-05 03:54:29 +1000 | [diff] [blame] | 282 | def _find_lteq(a, x): |
| 283 | 'Locate the leftmost value exactly equal to x' |
| 284 | i = bisect_left(a, x) |
| 285 | if i != len(a) and a[i] == x: |
| 286 | return i |
| 287 | raise ValueError |
| 288 | |
| 289 | |
| 290 | def _find_rteq(a, l, x): |
| 291 | 'Locate the rightmost value exactly equal to x' |
| 292 | i = bisect_right(a, x, lo=l) |
| 293 | if i != (len(a)+1) and a[i-1] == x: |
| 294 | return i-1 |
| 295 | raise ValueError |
| 296 | |
Steven D'Aprano | a474afd | 2016-08-09 12:49:01 +1000 | [diff] [blame^] | 297 | |
| 298 | def _fail_neg(values, errmsg='negative value'): |
| 299 | """Iterate over values, failing if any are less than zero.""" |
| 300 | for x in values: |
| 301 | if x < 0: |
| 302 | raise StatisticsError(errmsg) |
| 303 | yield x |
| 304 | |
| 305 | |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 306 | # === Measures of central tendency (averages) === |
| 307 | |
| 308 | def mean(data): |
| 309 | """Return the sample arithmetic mean of data. |
| 310 | |
| 311 | >>> mean([1, 2, 3, 4, 4]) |
| 312 | 2.8 |
| 313 | |
| 314 | >>> from fractions import Fraction as F |
| 315 | >>> mean([F(3, 7), F(1, 21), F(5, 3), F(1, 3)]) |
| 316 | Fraction(13, 21) |
| 317 | |
| 318 | >>> from decimal import Decimal as D |
| 319 | >>> mean([D("0.5"), D("0.75"), D("0.625"), D("0.375")]) |
| 320 | Decimal('0.5625') |
| 321 | |
| 322 | If ``data`` is empty, StatisticsError will be raised. |
| 323 | """ |
| 324 | if iter(data) is data: |
| 325 | data = list(data) |
| 326 | n = len(data) |
| 327 | if n < 1: |
| 328 | raise StatisticsError('mean requires at least one data point') |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 329 | T, total, count = _sum(data) |
| 330 | assert count == n |
| 331 | return _convert(total/n, T) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 332 | |
| 333 | |
Steven D'Aprano | a474afd | 2016-08-09 12:49:01 +1000 | [diff] [blame^] | 334 | def harmonic_mean(data): |
| 335 | """Return the harmonic mean of data. |
| 336 | |
| 337 | The harmonic mean, sometimes called the subcontrary mean, is the |
| 338 | reciprocal of the arithmetic mean of the reciprocals of the data, |
| 339 | and is often appropriate when averaging quantities which are rates |
| 340 | or ratios, for example speeds. Example: |
| 341 | |
| 342 | Suppose an investor purchases an equal value of shares in each of |
| 343 | three companies, with P/E (price/earning) ratios of 2.5, 3 and 10. |
| 344 | What is the average P/E ratio for the investor's portfolio? |
| 345 | |
| 346 | >>> harmonic_mean([2.5, 3, 10]) # For an equal investment portfolio. |
| 347 | 3.6 |
| 348 | |
| 349 | Using the arithmetic mean would give an average of about 5.167, which |
| 350 | is too high. |
| 351 | |
| 352 | If ``data`` is empty, or any element is less than zero, |
| 353 | ``harmonic_mean`` will raise ``StatisticsError``. |
| 354 | """ |
| 355 | # For a justification for using harmonic mean for P/E ratios, see |
| 356 | # http://fixthepitch.pellucid.com/comps-analysis-the-missing-harmony-of-summary-statistics/ |
| 357 | # http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2621087 |
| 358 | if iter(data) is data: |
| 359 | data = list(data) |
| 360 | errmsg = 'harmonic mean does not support negative values' |
| 361 | n = len(data) |
| 362 | if n < 1: |
| 363 | raise StatisticsError('harmonic_mean requires at least one data point') |
| 364 | elif n == 1: |
| 365 | x = data[0] |
| 366 | if isinstance(x, (numbers.Real, Decimal)): |
| 367 | if x < 0: |
| 368 | raise StatisticsError(errmsg) |
| 369 | return x |
| 370 | else: |
| 371 | raise TypeError('unsupported type') |
| 372 | try: |
| 373 | T, total, count = _sum(1/x for x in _fail_neg(data, errmsg)) |
| 374 | except ZeroDivisionError: |
| 375 | return 0 |
| 376 | assert count == n |
| 377 | return _convert(n/total, T) |
| 378 | |
| 379 | |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 380 | # FIXME: investigate ways to calculate medians without sorting? Quickselect? |
| 381 | def median(data): |
| 382 | """Return the median (middle value) of numeric data. |
| 383 | |
| 384 | When the number of data points is odd, return the middle data point. |
| 385 | When the number of data points is even, the median is interpolated by |
| 386 | taking the average of the two middle values: |
| 387 | |
| 388 | >>> median([1, 3, 5]) |
| 389 | 3 |
| 390 | >>> median([1, 3, 5, 7]) |
| 391 | 4.0 |
| 392 | |
| 393 | """ |
| 394 | data = sorted(data) |
| 395 | n = len(data) |
| 396 | if n == 0: |
| 397 | raise StatisticsError("no median for empty data") |
| 398 | if n%2 == 1: |
| 399 | return data[n//2] |
| 400 | else: |
| 401 | i = n//2 |
| 402 | return (data[i - 1] + data[i])/2 |
| 403 | |
| 404 | |
| 405 | def median_low(data): |
| 406 | """Return the low median of numeric data. |
| 407 | |
| 408 | When the number of data points is odd, the middle value is returned. |
| 409 | When it is even, the smaller of the two middle values is returned. |
| 410 | |
| 411 | >>> median_low([1, 3, 5]) |
| 412 | 3 |
| 413 | >>> median_low([1, 3, 5, 7]) |
| 414 | 3 |
| 415 | |
| 416 | """ |
| 417 | data = sorted(data) |
| 418 | n = len(data) |
| 419 | if n == 0: |
| 420 | raise StatisticsError("no median for empty data") |
| 421 | if n%2 == 1: |
| 422 | return data[n//2] |
| 423 | else: |
| 424 | return data[n//2 - 1] |
| 425 | |
| 426 | |
| 427 | def median_high(data): |
| 428 | """Return the high median of data. |
| 429 | |
| 430 | When the number of data points is odd, the middle value is returned. |
| 431 | When it is even, the larger of the two middle values is returned. |
| 432 | |
| 433 | >>> median_high([1, 3, 5]) |
| 434 | 3 |
| 435 | >>> median_high([1, 3, 5, 7]) |
| 436 | 5 |
| 437 | |
| 438 | """ |
| 439 | data = sorted(data) |
| 440 | n = len(data) |
| 441 | if n == 0: |
| 442 | raise StatisticsError("no median for empty data") |
| 443 | return data[n//2] |
| 444 | |
| 445 | |
| 446 | def median_grouped(data, interval=1): |
Zachary Ware | df2660e | 2015-10-27 22:00:41 -0500 | [diff] [blame] | 447 | """Return the 50th percentile (median) of grouped continuous data. |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 448 | |
| 449 | >>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5]) |
| 450 | 3.7 |
| 451 | >>> median_grouped([52, 52, 53, 54]) |
| 452 | 52.5 |
| 453 | |
| 454 | This calculates the median as the 50th percentile, and should be |
| 455 | used when your data is continuous and grouped. In the above example, |
| 456 | the values 1, 2, 3, etc. actually represent the midpoint of classes |
| 457 | 0.5-1.5, 1.5-2.5, 2.5-3.5, etc. The middle value falls somewhere in |
| 458 | class 3.5-4.5, and interpolation is used to estimate it. |
| 459 | |
| 460 | Optional argument ``interval`` represents the class interval, and |
| 461 | defaults to 1. Changing the class interval naturally will change the |
| 462 | interpolated 50th percentile value: |
| 463 | |
| 464 | >>> median_grouped([1, 3, 3, 5, 7], interval=1) |
| 465 | 3.25 |
| 466 | >>> median_grouped([1, 3, 3, 5, 7], interval=2) |
| 467 | 3.5 |
| 468 | |
| 469 | This function does not check whether the data points are at least |
| 470 | ``interval`` apart. |
| 471 | """ |
| 472 | data = sorted(data) |
| 473 | n = len(data) |
| 474 | if n == 0: |
| 475 | raise StatisticsError("no median for empty data") |
| 476 | elif n == 1: |
| 477 | return data[0] |
| 478 | # Find the value at the midpoint. Remember this corresponds to the |
| 479 | # centre of the class interval. |
| 480 | x = data[n//2] |
| 481 | for obj in (x, interval): |
| 482 | if isinstance(obj, (str, bytes)): |
| 483 | raise TypeError('expected number but got %r' % obj) |
| 484 | try: |
| 485 | L = x - interval/2 # The lower limit of the median interval. |
| 486 | except TypeError: |
| 487 | # Mixed type. For now we just coerce to float. |
| 488 | L = float(x) - float(interval)/2 |
Steven D'Aprano | 3b06e24 | 2016-05-05 03:54:29 +1000 | [diff] [blame] | 489 | |
| 490 | # Uses bisection search to search for x in data with log(n) time complexity |
Martin Panter | f157982 | 2016-05-26 06:03:33 +0000 | [diff] [blame] | 491 | # Find the position of leftmost occurrence of x in data |
Steven D'Aprano | 3b06e24 | 2016-05-05 03:54:29 +1000 | [diff] [blame] | 492 | l1 = _find_lteq(data, x) |
Martin Panter | f157982 | 2016-05-26 06:03:33 +0000 | [diff] [blame] | 493 | # Find the position of rightmost occurrence of x in data[l1...len(data)] |
Steven D'Aprano | 3b06e24 | 2016-05-05 03:54:29 +1000 | [diff] [blame] | 494 | # Assuming always l1 <= l2 |
| 495 | l2 = _find_rteq(data, l1, x) |
| 496 | cf = l1 |
| 497 | f = l2 - l1 + 1 |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 498 | return L + interval*(n/2 - cf)/f |
| 499 | |
| 500 | |
| 501 | def mode(data): |
| 502 | """Return the most common data point from discrete or nominal data. |
| 503 | |
| 504 | ``mode`` assumes discrete data, and returns a single value. This is the |
| 505 | standard treatment of the mode as commonly taught in schools: |
| 506 | |
| 507 | >>> mode([1, 1, 2, 3, 3, 3, 3, 4]) |
| 508 | 3 |
| 509 | |
| 510 | This also works with nominal (non-numeric) data: |
| 511 | |
| 512 | >>> mode(["red", "blue", "blue", "red", "green", "red", "red"]) |
| 513 | 'red' |
| 514 | |
| 515 | If there is not exactly one most common value, ``mode`` will raise |
| 516 | StatisticsError. |
| 517 | """ |
| 518 | # Generate a table of sorted (value, frequency) pairs. |
| 519 | table = _counts(data) |
| 520 | if len(table) == 1: |
| 521 | return table[0][0] |
| 522 | elif table: |
| 523 | raise StatisticsError( |
| 524 | 'no unique mode; found %d equally common values' % len(table) |
| 525 | ) |
| 526 | else: |
| 527 | raise StatisticsError('no mode for empty data') |
| 528 | |
| 529 | |
| 530 | # === Measures of spread === |
| 531 | |
| 532 | # See http://mathworld.wolfram.com/Variance.html |
| 533 | # http://mathworld.wolfram.com/SampleVariance.html |
| 534 | # http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance |
| 535 | # |
| 536 | # Under no circumstances use the so-called "computational formula for |
| 537 | # variance", as that is only suitable for hand calculations with a small |
| 538 | # amount of low-precision data. It has terrible numeric properties. |
| 539 | # |
| 540 | # See a comparison of three computational methods here: |
| 541 | # http://www.johndcook.com/blog/2008/09/26/comparing-three-methods-of-computing-standard-deviation/ |
| 542 | |
| 543 | def _ss(data, c=None): |
| 544 | """Return sum of square deviations of sequence data. |
| 545 | |
| 546 | If ``c`` is None, the mean is calculated in one pass, and the deviations |
| 547 | from the mean are calculated in a second pass. Otherwise, deviations are |
| 548 | calculated from ``c`` as given. Use the second case with care, as it can |
| 549 | lead to garbage results. |
| 550 | """ |
| 551 | if c is None: |
| 552 | c = mean(data) |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 553 | T, total, count = _sum((x-c)**2 for x in data) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 554 | # The following sum should mathematically equal zero, but due to rounding |
| 555 | # error may not. |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 556 | U, total2, count2 = _sum((x-c) for x in data) |
| 557 | assert T == U and count == count2 |
| 558 | total -= total2**2/len(data) |
| 559 | assert not total < 0, 'negative sum of square deviations: %f' % total |
| 560 | return (T, total) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 561 | |
| 562 | |
| 563 | def variance(data, xbar=None): |
| 564 | """Return the sample variance of data. |
| 565 | |
| 566 | data should be an iterable of Real-valued numbers, with at least two |
| 567 | values. The optional argument xbar, if given, should be the mean of |
| 568 | the data. If it is missing or None, the mean is automatically calculated. |
| 569 | |
| 570 | Use this function when your data is a sample from a population. To |
| 571 | calculate the variance from the entire population, see ``pvariance``. |
| 572 | |
| 573 | Examples: |
| 574 | |
| 575 | >>> data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5] |
| 576 | >>> variance(data) |
| 577 | 1.3720238095238095 |
| 578 | |
| 579 | If you have already calculated the mean of your data, you can pass it as |
| 580 | the optional second argument ``xbar`` to avoid recalculating it: |
| 581 | |
| 582 | >>> m = mean(data) |
| 583 | >>> variance(data, m) |
| 584 | 1.3720238095238095 |
| 585 | |
| 586 | This function does not check that ``xbar`` is actually the mean of |
| 587 | ``data``. Giving arbitrary values for ``xbar`` may lead to invalid or |
| 588 | impossible results. |
| 589 | |
| 590 | Decimals and Fractions are supported: |
| 591 | |
| 592 | >>> from decimal import Decimal as D |
| 593 | >>> variance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")]) |
| 594 | Decimal('31.01875') |
| 595 | |
| 596 | >>> from fractions import Fraction as F |
| 597 | >>> variance([F(1, 6), F(1, 2), F(5, 3)]) |
| 598 | Fraction(67, 108) |
| 599 | |
| 600 | """ |
| 601 | if iter(data) is data: |
| 602 | data = list(data) |
| 603 | n = len(data) |
| 604 | if n < 2: |
| 605 | raise StatisticsError('variance requires at least two data points') |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 606 | T, ss = _ss(data, xbar) |
| 607 | return _convert(ss/(n-1), T) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 608 | |
| 609 | |
| 610 | def pvariance(data, mu=None): |
| 611 | """Return the population variance of ``data``. |
| 612 | |
| 613 | data should be an iterable of Real-valued numbers, with at least one |
| 614 | value. The optional argument mu, if given, should be the mean of |
| 615 | the data. If it is missing or None, the mean is automatically calculated. |
| 616 | |
| 617 | Use this function to calculate the variance from the entire population. |
| 618 | To estimate the variance from a sample, the ``variance`` function is |
| 619 | usually a better choice. |
| 620 | |
| 621 | Examples: |
| 622 | |
| 623 | >>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25] |
| 624 | >>> pvariance(data) |
| 625 | 1.25 |
| 626 | |
| 627 | If you have already calculated the mean of the data, you can pass it as |
| 628 | the optional second argument to avoid recalculating it: |
| 629 | |
| 630 | >>> mu = mean(data) |
| 631 | >>> pvariance(data, mu) |
| 632 | 1.25 |
| 633 | |
| 634 | This function does not check that ``mu`` is actually the mean of ``data``. |
| 635 | Giving arbitrary values for ``mu`` may lead to invalid or impossible |
| 636 | results. |
| 637 | |
| 638 | Decimals and Fractions are supported: |
| 639 | |
| 640 | >>> from decimal import Decimal as D |
| 641 | >>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")]) |
| 642 | Decimal('24.815') |
| 643 | |
| 644 | >>> from fractions import Fraction as F |
| 645 | >>> pvariance([F(1, 4), F(5, 4), F(1, 2)]) |
| 646 | Fraction(13, 72) |
| 647 | |
| 648 | """ |
| 649 | if iter(data) is data: |
| 650 | data = list(data) |
| 651 | n = len(data) |
| 652 | if n < 1: |
| 653 | raise StatisticsError('pvariance requires at least one data point') |
Steven D'Aprano | b28c327 | 2015-12-01 19:59:53 +1100 | [diff] [blame] | 654 | T, ss = _ss(data, mu) |
| 655 | return _convert(ss/n, T) |
Larry Hastings | f5e987b | 2013-10-19 11:50:09 -0700 | [diff] [blame] | 656 | |
| 657 | |
| 658 | def stdev(data, xbar=None): |
| 659 | """Return the square root of the sample variance. |
| 660 | |
| 661 | See ``variance`` for arguments and other details. |
| 662 | |
| 663 | >>> stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75]) |
| 664 | 1.0810874155219827 |
| 665 | |
| 666 | """ |
| 667 | var = variance(data, xbar) |
| 668 | try: |
| 669 | return var.sqrt() |
| 670 | except AttributeError: |
| 671 | return math.sqrt(var) |
| 672 | |
| 673 | |
| 674 | def pstdev(data, mu=None): |
| 675 | """Return the square root of the population variance. |
| 676 | |
| 677 | See ``pvariance`` for arguments and other details. |
| 678 | |
| 679 | >>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75]) |
| 680 | 0.986893273527251 |
| 681 | |
| 682 | """ |
| 683 | var = pvariance(data, mu) |
| 684 | try: |
| 685 | return var.sqrt() |
| 686 | except AttributeError: |
| 687 | return math.sqrt(var) |