solutions {uwedragon}

Find individual sample values from the sample mean and standard deviation

Description

For integer based scales, finds possible solutions for each value within a sample. This is revealed upon providing sample size, minimum possible value, maximum possible value, mean, standard deviation (and optionally median).

Usage

solutions(
  n,
  min_poss,
  max_poss,
  usermean,
  usersd,
  meandp = NULL,
  sddp = NULL,
  usermed = NULL
)

Arguments

n

Sample size.

min_poss

Minimum possible value. If sample minimum is disclosed, this can be inserted here, otherwise use the theoretical minimum. If there is no theoretical maximum 'Inf' can be inserted.

max_poss

Maximum possible value. If sample maximum is disclosed, this can be inserted here, otherwise use the theoretical maximum. If there is no theoretical minimum '-Inf' can be inserted.

usermean

Sample mean.

usersd

Sample standard deviation, i.e. n-1 denominator.

meandp

(optional, default=NULL) Number of decimal places mean is reported to, only required if including trailing zeroes.

sddp

(optional, default=NULL) Number of decimal places standard deviation is reported to, only required if including trailing zeroes.

usermed

(optional, default=NULL) Sample median.

Details

For use with data measured on a scale with 1 unit increments. Samuelson's inequality [1] used to further restrict the minimum and maximum. All possible combinations within this inequality are calculated [2] for factorial(n+k-1)/(factorial(k)*factorial(n-1))<65,000,000.

No restriction on number of decimal places input. Reporting less than two decimal places will reduce the chances of unique solution to all sample values being uncovered [3]

Additional options to specify number of digits following the decimal place that are reported, required for trailing zeroes.

Value

Outputs possible combinations of original integer sample values.

References

[1] Samuelson, P.A, 1968, How deviant can you be? Journal of the American Statistical Association, Vol 63, 1522-1525.

[2] Allenby, R.B. and Slomson, A., 2010. How to count: An introduction to combinatorics. Chapman and Hall/CRC.

[3] Derrick, B., Green, L., Kember, K., Ritchie, F. & White P, 2022, Safety in numbers: Minimum thresholding, Maximum bounds, and Little White Lies. Scottish Economic Society Annual Conference, University of Glasgow, 25th-27th April 2022

Examples

# EXAMPLE 1
# Seven observations are taken from a five-point Likert scale (coded 1 to 5).
# The reported mean is 2.857 and the reported standard deviation is 1.574.

solutions(7,1,5,2.857,1.574)

# For this mean and standard deviation there are two possible distributions:
# 1  1  2  3  4  4  5
# 1  2  2  2  3  5  5

# Optionally adding median value of 3.

solutions(7,1,5,2.857,1.574, usermed=3)

# uniquely reveals the raw sample values:
# 1  1  2  3  4  4  5


# EXAMPLE 2
# The mean is '4.00'.
# The standard deviation is '2.00'.
# Narrower set of solutions found specifying 2dp including trailing zeroes.

solutions(3,-Inf,Inf,4.00,2.00,2,2)

# uniquely reveals the raw sample values:
# 2  4  6