This page describes the relationship between sample size and error in estimating Standard Deviations (SD) from a sample

As most statistical calculations are based on the critical assumption that the SD of a measurement is known and valid, the precision of an estimated SD is therefore important.

In many cases in the clinical situation, an assumed value for SD is used, based on published figures or from estimates using a small pilot study, but this often results in misleading conclusions

In particular, in engineering and in high precision biochemical laboratories, a valid assumed SD is critical, and the calculations presented in this panel is one of the methods used to obtain this.

The error, the confidence interval of the SD estimated, is expressed as a percent of the value. The greater the sample size, the narrower would be this confidence interval.

Two calculations are offered in this panel

• At the planning stage of the study, to estimate the sample size required to establish a desired confidence interval.
• After collecting all the data, to estimate the SD and its confidence interval

The other sub-panels for SD are

• Javascript program to calculate sample size, and to estimate confidence intervals
• R codes to do the same
• Tables of sample size and confidence intervals in the commonly used range of values

References

Greenwood JA and Sandomire MM (1950) Journal of the American Statistical Association 45 (250) p. 257 - 260

Burnett RW (1975)Accurate estimation of standard deviations for quantitative methods used in clinical chemistry. Clin. Chem. 21 (13) p. 1935-1938

This panel presents 2 tables related to sample size and Standard Deviations. The first estimates the sample size required at the planning stage, and the second the confidence interval from the data obtained

Table of Sample size

• Error is the margin of error in terms of the percentage of the Standard Deviation found (SD), so that the confidence interval is SD ± error/100 x SD
• 80, 90, 95, and 99 percent confidence of ± 1% to 50% od SD found are provided in this table
• The cells contain the sample size required
• Example: If we required the 95% confidence interval of the Standard Deviation found to be ±10% of its value, then we need to calculate the Standard Deviation with a sample of 193

Error	Percent Confidence				Error	Percent Confidence				Error	Percent Confidence
	80%	90%	95%	99%		80%	90%	95%	99%		80%	90%	95%	99%
±1%	8213	13529	19209	33175	±2%	2054	3383	4803	8295	±3%	914	1504	2135	3687
±4%	515	847	1202	2074	±5%	330	542	770	1328	±6%	230	377	535	923
±7%	169	277	393	678	±8%	130	213	301	519	±9%	103	168	238	411
±10%	84	137	193	333	±11%	69	113	160	275	±12%	59	95	135	231
±13%	50	81	115	197	±14%	43	70	99	170	±15%	38	61	87	149
±16%	34	54	76	131	±17%	30	48	68	116	±18%	27	43	60	103
±19%	24	39	54	93	±20%	22	35	49	84	±21%	20	32	45	76
±22%	18	29	41	70	±23%	17	27	37	64	±24%	16	25	35	59
±25%	15	23	32	54	±26%	14	21	30	50	±27%	13	20	28	46
±28%	12	19	26	43	±29%	11	17	24	40	±30%	11	16	23	38
±31%	10	15	21	35	±32%	10	15	20	33	±33%	9	14	19	31
±34%	9	13	18	30	±35%	8	12	17	28	±36%	8	12	16	27
±37%	7	11	15	25	±38%	7	11	14	24	±39%	7	10	14	23
±40%	7	10	13	22	±41%	6	9	13	21	±42%	6	9	12	20
±43%	6	9	12	19	±44%	6	8	11	18	±45%	6	8	11	17
±46%	5	8	10	17	±47%	5	7	10	16	±48%	5	7	9	15
±49%	5	7	9	15	±50%	5	7	9	14

Sample Size for SD

Each row a calculation   Data in 2 columns     - Col 1 = % confidence     - Col 2 = width of interval as percent of SD found     - e.g.: 95 10 means 95% CI = ±10% of SD found

Confidence Interval for SD

Each row a calculation   Data in 2 columns     - Col 1 = % confidence     - Col 2 = sample size of data     - e.g.: 95 270 means 95% CI ±proportion od SD) with sample size of 270

This sub-panel provides the R codes for estimating sample size needed and error estimated with Standard Deviation, algorithm is as described in

Burnett RW (1975) Accurate estimation of standard deviations for quantitative methods used in clinical chemistry. Clin. Chem. 21 (13) p. 1935-1938

Section 1. Supportive subroutines

ChiSqToP<-function(chi,degF)      #function to calculate probability from chisq and df
{
  return (1 - pchisq(chi, df=degF))  #probability
}

Confidence <- function (u,df)         # common to both ssiz and error
{
  #x1 = pow(1 - u,2) * df
  #x2 = pow(1 + u,2) * df  
  x1 = (1 - u)^2 * df
  x2 = (1 + u)^2 * df  
  return (ChiSqToP(x1,df) - ChiSqToP(x2,df));
}

# Calculate sample size from confidence and % error

SSizSD <- function (Cf,Er)    # percent confidence, tolerable error (as % of SD found)
{
  Cf = Cf / 100;                  # converts to probability
  Er = Er / 100;                  # converts to probability
  # iterate for correct sample size
  nl = 0;
  nr = 100000000;
  nm = nr / 2;
  p = Confidence(Er,round(nm)); 
  while(abs(nl-nr)>1)
  {
    if(p<Cf) { nl = nm; } else {  nr = nm; }
    nm = (nr + nl) / 2.0;  
    p = Confidence(Er,round(nm));
  }
  #print(ceiling(nm) + 1)
  return (ceiling(nm) + 1);    # nm is df so sample size needs plus 1    
}

# Calculate Error from sample size (n) and % confidence

ErrSD <- function(cf,ssiz)               # n= sample size, Cf = percent confidence
{
  cf = cf / 100;                   # converts to probability
  df = ssiz-1;                        # degrees of confidence
  # iterate for error
  ul = 0;
  ur = 1;
  um = 0.5;
  p = Confidence(um,df);
  while(abs(ul-ur)>0.0001)
  {
    if(p>cf) { ur = um;  } else  { ul = um; }
    um = (ul + ur) / 2.0;
    p = Confidence(um,df);
  }
  return (um * 100);
}

Section 2. Main Programs

Propgram 1. Sample size

 
txt = ("
Cf  Err
90   10
95    5
99    1
")                        # input data Cf=% confidence Err= error in % of SD                   
df <- read.table(textConnection(txt),header=TRUE)      
#df                        # optional display of input data frame

# extract arrays from data frame
arCf <- df$Cf
arErr <- df$Err
# Create results array
arSSiz <- vector()
# Calculate
for(i in 1:nrow(df))
{
  cf = arCf[i]
  er = arErr[i]
  arSSiz <- append(arSSiz,SSizSD(cf, er)) # append sample size to result array
}
# Incorporatw results to original data frame
df$SSiz <- arSSiz
# show data frame with results
df

The results are as follows

> df
  Cf Err  SSiz
1 90  10   137
2 95   5   769
3 99   1 33175

Propgram 2. Error as % of SD found

 
txt = ("
Cf  SSiz
90   137
95   769
99 33175
")        # input data Cf=% confidence, SSiz = sample size                  
df <- read.table(textConnection(txt),header=TRUE)      
#df                        # optional display of input data frame

# extract arrays from data frame
arCf <- df$Cf
arSSiz <- df$SSiz
# Create results array
arEr <- vector()
# Calculate
for(i in 1:nrow(df))
{
  cf = arCf[i]
  ssiz = arSSiz[i]
  print(c(cf,ssiz))
  arEr <- append(arEr,ErrSD(cf, ssiz)) # append error to result array
}
# Incorporatw results to original data frame
df$Error <- arEr
# show data frame with results
df

The results are

> df
  Cf  SSiz     Error
1 90   137 9.9700928
2 95   769 5.0018311
3 99 33175 0.9979248