Does Dividing the Range by Four Provide an Accurate Estimate of a Standard Deviation in Family Science Research? A Teaching Editorial

ABSTRACT

Occasionally, scientific reports have omitted information on standard deviations, making estimates of effect sizes very difficult to impossible. In such situations, several scholars have recommended obtaining an estimate of the standard deviation of distributions by dividing the range of the distribution (highest value minus lowest value) by four. However, there appears to be little evidence to confirm the validity of this approach. Articles from 2012 to 2015 in the journal Marriage & Family Review were surveyed to find instances where demographic variables (age, education, duration of relationship, number of children) were reported with both standard deviations and ranges. Ratios between range and standard deviations were calculated by several rules of thumb or more complex formulas and compared with the actual ratios obtained. Results indicated that dividing by five in general provided a more accurate estimate of actual standard deviations but accuracy in predicting the true ratio between range and standard deviation was substantially related to the position of the mean score within the range of scores with larger divisors needed as the mean approached either the minimum or the maximum values of the demographic variable (skew). Other recent formulae for estimating the standard deviation were also evaluated, but the skew-based approach appeared to be more accurate than the others. However, further investigation in other samples is needed because the skew-based approach was derived from observation of the data here, which might not replicate in different sets of data.

KEYWORDS:

• demographic variables
• estimation of standard deviations
• meta-analyses
• research methodology
• reviews of the literature

Introduction

Not all scholars report standard deviations for their demographic or other variables. Although such omissions might seem minor in many situations, in some situations effect sizes cannot be calculated without knowing the standard deviations as well as the mean scores. In some situations, one may obtain substantial effect sizes even without statistical significance (p < .05) at conventional levels. Thus, omission of standard deviations can lead to incorrect conclusions, such as effects being deemed nonsignificant and not substantial, when effect sizes would have been substantial had they been able to be calculated from means and standard deviations of comparative groups (Schumm, 2014). It is also possible to obtain statistically significant results from de facto trivial effect sizes, but without the ability to calculate those effect sizes, one might conclude that effects were significant (correct) and substantial (incorrect). Either way, misleading results can be presented if effect sizes cannot be determined in addition to levels of statistical significance.

As noted by Wiebe et al. (2006), the results of meta-analyses can be biased when effect sizes cannot be determined because of omission of information about variances. Wiebe et al. recommend asking authors for information on unreported variances, but what are scholars to do if standard deviations are not provided and authors refuse to release them when requested? For example, Kwon and Reis (2015) state that “not all studies report these quantities [standard deviations or variances] directly” (p. 1). In addition to finding that many studies did not report information on effect sizes or standard deviations, Wiebe et al. found that there was a clear lack of standardization in methods for dealing with such problems. In addition, some authors have confused the terms standard error, standard deviation, and variance (Norris, Lau, Smith, Schmid, & Engelgau, 2002, p. 1161), so interpretation of their data becomes problematic. In such instances, the approaches below might help solve the confusion.

Prior solutions

Brase and Brase’s solution: Range divided by four

Brase and Brase (2015), in their basic statistics text, have offered one solution to this dilemma. Brase and Brase (2015) report that “for a symmetrical and bell-shaped distribution, approximately 95% of the data lies within two standard deviations of the mean. Therefore, a 95% range of data values extending from μ−2σ to μ+2σ is often used for ‘commonly occurring’ data values. Note that the interval from μ−2σ to μ+2 σ is 4σ in length. This leads to a ‘rule of thumb’ for estimating the standard deviation from a 95% range of values” (p. 311) in which the standard deviation is estimated by dividing the difference between the highest and lowest values of the variable by 4. Fleissig (2014) also stated the same rule: “For a normal distribution, the standard deviation can be approximated by the range divided by four” (p. 46). Likewise, McClendon, Politzer, Christian, and Fernandez (1997) reported that they calculated “an estimate of the standard deviation (SD) for each distribution at the range divided by four” (p. 235). Wan, Wang, Liu, and Tong (2014) cited dividing the range by four as a “well-known rule of thumb” (p. 3), which can be traced back at least as far as Mendenhall, Ott, and Scheaffer (1971) or Scheaffer, Mendenhall, and Ott (1979, p. 7). When interquartiles are available, a related rule of thumb is to divide the interquartile range by 1.35 (Kwon & Reis, 2015, p. 2).

An alternative solution: Range divided by six

However, Chebyshev’s theorem indicates that for any set of data, including data that are not normally distributed, 88.9% of the data will fall in a 6σ interval and 93.8% of the data will fall in an 8σ interval (Brase & Brase, 2015, p. 111). Thus, for non-normal distributions, it might be more appropriate to divide by six to estimate a standard deviation from a known range, an approach recommended by Hozo et al. (2005) when data are not normally distributed. Norris et al. (2002, p. 1161) recommended dividing by 5.88 rather than 6, citing Falkenberg, Elwing, Goransson, Hellstrand, and Riis (1986). Pearson (1932, p. 416) developed a table of values where (for samples drawn from a normally distributed population) the ratio of the lower and upper .5% of the range to the standard deviation ranged from 4.64 (n = 2) to 5.19 (n = 100), the ratio increasing with larger sample sizes. However, highly skewed data may not work well with traditional rules of thumb or more complicated methods of estimation, even if non-normal distributions are assumed (Bland, 2015; Kwon & Reis, 2015, p. 8).

More complex solutions for estimating standard deviations

Recognizing the limitations of various rules of thumb for estimating standard deviations, Hozo, Djulbegovic, and Hozo (2005) offered an improved solution. For samples of 15 or less, they recommended their formula that used the minimum and maximum values as well as the median (p. 4, no. 16), even though they assumed a large sample size in developing that formula from more complex formulae (no. 12, p. 3, and no. 15, p. 4). For data from symmetric distributions, their more complex formula simplified to the range divided by the square root of 12 (= 3.46). For samples between 16 and 70, they preferred dividing the range by four. For larger samples, they preferred dividing the range by six. However, Bland (2015) has found, through simulation studies, that their estimates for standard deviations tend to be too large, that is, larger than the actual standard deviations from the original data, especially for samples with n > 30. Bland found that for samples of size 100 and 500, the standard deviation was overestimated by nearly 20% and 30%, respectively.

More recently, Wan et al. (2014) offered an improvement to Hozo et al.’s (2005) recommendations. Wan et al. provided a table (p. 4) that specified a divisor to the range, a divisor that varied depending on the sample size, from n = 1 to n = 50. In summary, rounding their numbers, for n = 9, divide by 3; for n = 16, divide by 3.5; for n = 27, divide by 4.0; for n = 50, divide by 4.5. They found the point that best fits dividing by six occurred for a sample size of about 463. Some of their better estimation procedures require knowledge of the interquartile range, which is seldom provided in social science research reports.

Study objective

The overall objective of this report was to assess the validity of various rules for estimating standard deviations from the range of a variable by looking at several demographic variables published in the journal Marriage & Family Review between 2012 and 2015, using data where the authors had provided means, standard deviations, and ranges for those variables. We considered only basic approaches to estimating standard deviations from ranges; more complex procedures can be reviewed elsewhere (Kwon & Reis, 2015). We do not consider here the variety of approaches available for estimating mean scores from other parameters.

Methods

The specific objective of this study was to compare known standard deviations from a wide variety of quantitative and qualitative articles with estimated standard deviations deduced by dividing variable ranges by four or, possibly, more accurate numbers, such as five, six, or eight. Although qualitative reports often do not report characteristics of social-psychological variables, they often do report demographic information. In some cases means, standard deviations, and ranges were calculated from raw data provided within an article, even if not provided per se by the authors. To achieve consistency across variables with widely varying standard deviations, we focused on the actual or estimated ratios between range and divisor(s) that would yield the most accurate estimate of standard deviations.

Sample

Articles published in Marriage & Family Review between 2012 and 2015 were reviewed to locate both demographic characteristics for which authors had reported means, standard deviations, and ranges. The demographic characteristics included age, duration of relationship, number of children, and education. Data from both quantitative and qualitative articles were used.

Articles included

Our analysis required that means, standard deviations, and ranges be included for each variable to be included in our analysis. Some articles (e.g., Dhar, 2014, p. 539; Lesch & Scheffler, 2015) did not report means, standard deviations, or ranges altogether but provided raw data from which any missing items could be calculated. In some cases means, standard deviations, or ranges were reported in different (page) locations within an article (e.g., Chaney, Mitchell, & Barker, 2014, pp. 566 and 568) but could still be used for our analyses. It was not uncommon for authors to report data on means, standard deviations, and ranges for some demographic variables but not for other demographic variables (e.g., Chaney, 2012; Cohen & Finzi-Dottan, 2014, p. 399; Mortensen & Barnett, 2015, p. 6), limiting their inclusion to only those demographic variables that did feature means, standard deviations, and ranges. However, a number of articles had to be excluded from the analysis, as discussed below.

Articles excluded from analysis

Some studies were not included in the following analyses for several reasons. First, studies that reported data for fewer than 10 participants were excluded from analysis (e.g., Nurullah, 2013; Skogrand, Mendez, & Higginbotham, 2013, reported data on only four participants; Allen and Mitchell (2015) relied on only six participants; Allen, 2012; Cohen, 2013; Hoser, 2012, reported data on only eight participants in each of their studies; Stambulich, Pooley, Gately, & Taylor, 2012, reported on only nine participants). Second, cases in which ranges were severely restricted were excluded (e.g., Dhar, 2013a, p. 6, duration of marriage between only 6 months and 2 years).

Third, in some articles, the sampling frame was programs or something other than persons for whom descriptives such as age, education, relationship duration, or number of children, would be appropriate (e.g., Doty & Dworkin, 2014; Jagannathan, 2012; Myers-Bowman & Jurich, 2015; Otters & Hollander, 2015; Ozgur & Aydin, 2012; Rosenblatt & Li, 2012). Likewise, some articles were reviews of the literature with respect to either research (e.g., Allen, 2015; Cameron & Cameron, 2012; Chinuba, 2015; Edwards, 2012; Jensen & Howard, 2015; Nielsen, 2014; Orengo-Aguayo, 2015; Schumm, 2012a, 2012b), development of a research methodology (Wood, Gnonhosou, & Bowling, 2015), theory or theory development (Hall, 2012; Martins, McNamee, & Guanaes-Lorenzi, 2014; Ruppel, 2015), family life education programs (Childs & Duncan, 2012; Hawkins & Ooms, 2012; Lewton & Nievar, 2012; Reck, Higginbotham, Skogrand, & Davis, 2012; Schrannn & Gomez-Scott, 2012; Whitton & Buzzella, 2012), or clinical/ethical issues (Bailey, 2012; Rosik, 2014) and would not normally be expected to have included demographic data on participants. In addition, some program evaluation reports did not specify demographic data on the participants (Barden et al., 2015).

Fourth, many studies omitted data on standard deviations (e.g., Andrade & Mikula, 2014, p. 293; Bailey, Haynes, & Letiecq, 2013, p. 679; Carlson et al., 2014, p. 83; Cohen, 2013, p. 495; Desrochers, Sargent, & Hostetler, 2012; Galloway, Engstrom, & Emmers-Sommer, 2015, p. 695; Goodman, Dollahite, & Marks, 2012; Harris, Schramm, Marshall, & Lee, 2012, p. 398, age at marriage; Hurt, 2014, p. 453; Kushner, Pitre, Williamson, Breitkreuz, & Rempel, 2014, p. 10, marriage duration; Leerkes, Supple, & Gudmunson, 2014, p. 439; Mercadante, Taylor, & Pooley, 2014, p. 323; Nesteruk & Gramescu, 2012, p. 46; Nesteruk, Helmstetter, Gramescu, Siyam, & Price, 2015, p. 472; Perry, Cavanaugh, Dunbar, & Leerkes, 2015, p. 234; Polachek & Wallace, 2015, p. 212; Price & Nesteruk, 2015, p. 422; Shalev, Baum, & Itzhaky, 2012, p. 215; Sibley, Springer, Vennum, & Hollist, 2015, p. 186; Vaterlaus, Beckert, Tulane, & Bird, 2014, p. 695; Verma & Satayanarayana, 2013, p. 739; Wright & Brandt, 2015, p. 549), whereas others omitted data on ranges (e.g., Archuleta, 2013, p. 400; Bernstein, Keltner, & Laurent, 2012, p. 717; Davis, Jacobsen, & Anderson, 2012, p. 611; Emmers-Sommer, Hertlein, & Kennedy, 2013, p. 353; Jurich, Leite, & Chabot, 2015, p. 739; Keeling, Wessely, Dandeker, Jones, & Fear, 2015, p. 280; Kleftaras & Alexopoulos, 2015, p. 568; McAllister, Duncan, & Busby, 2013, p. 569; Merrifield, Lucero-Liu, & Gamble, 2014, p. 511; Negy, Pearte, & Lacefield, 2013, p. 59; Parade, Supple, & Helms, 2012, p. 156; Pedersen & Minnotte, 2012, p. 278; Rodriguez & Adamsons, 2012, p. 257; Shafer, 2013, p. 113; Stanley, 2012, p. 589; Todesco, 2015, p. 71; Willoughby, Madsen, Carroll, & Busby, 2015, p. 594; Woszidlo & Segrin, 2013, p. 528), rendering our statistical tests impossible, lacking the necessary data.

Fifth, some studies provided only ranges for key variables (Bates & Goodsell, 2013, p. 31; Burr et al., 2013, p. 292, length of marriage; Wright, 2013, pp. 314 and 318), whereas others provided only means (Aghajanian & Thompson, 2013, p. 127; Cigala, Fruggeri, & Venturelli, 2013, p. 721; Dhar, 2013b, p. 270; Dollahite, Hawkins, & Parr, 2012, p. 342; Forste & Jacobsen, 2013, p. 335).

Sixth, some authors did not report information on certain demographic variables in terms of means, standard deviations, or ranges (e.g., age, Bohm & Shapley, 2013; Bowers, Wiley, Jones, Ogolsky, & Branscomb, 2014; Bradford, Higginbotham, & Skogrand, 2014; Cassidy, Lawrence, Vierbuchen, & Konold, 2013, p. 198; Donovan & Emmers-Sommer, 2012; Golub, Reid, Strickler, & Dunlap, 2013; Sahoo, 2013; Zabriskie & Ward, 2013). For example, Muraco and Curran (2012, p. 232) presented data on age and education in terms of medians and ranges rather than means or standard deviations. For example, educational attainment and even age were often presented in terms of percentages of high school or college graduates rather than in terms of specific years of formal education (e.g., Baydoun, 2015; Gustafson & Fransson, 2015; Knabb & Pelletier, 2013; Moen, Bradford, Lee, Harris, & Stewart, 2015); age was sometimes presented in terms of percent of respondents within various decades of life (e.g., Golub et al., 2013, p. 633; Harris et al., 2012, p. 394).

Description of data

Overall, there were 67 sets of data from 51 articles. A higher percentage of articles were from 2013 (46.3%) than from 2012 (14.9%), 2014 (26.9%) or 2015 (11.9%). Sample sizes ranged from 12 to 6,391 with an average of 727.4 and a median of 362.

Tables 1 through 4 provide raw data for age, duration of relationship, number of children, and education. Age, duration of relationship, and education were only evaluated if specified in years; when specified in months, age or duration of relationship were recoded into years by dividing by 12 months. Two possible errors concerning published standard deviations were detected in the data, and those values were not included in our analyses (see footnote for Table 1).

Table 1. Descriptive data for age of respondents or ages of respondents’ siblings compared with range divided by four estimates for standard deviations from 35 articles published in Marriage & Family Review, 2012–2015.

Study authors, date of publication, page number(s) containing data	Type of study	N	Mean	SD	Range
Dhar (2013a), p. 6	Qualitative	30	20.20	1.56	18–24
Squire, Limke, and Jones (2013), p. 139	Quantitative	356	19.77	1.91	18–25
Thompson, Smith, Whiting, and Bradford (2013), p. 153	Qualitative	20	37.00	6.22	30–50
Higginbotham and Agee (2013), p. 181	Quantitative	447	42.28	14.6	19–92
			46.00	15.5	18–96
Knabb and Pelletier (2013), p. 236	Quantitative	178	18.88	1.01	17–24
Milevsky and Heerwagen (2013), p. 254	Qualitative	52	19.85	1.16	18–23
		49	19.82	1.17	18–23
Burr, Hubler, Larzelere, and Gardner (2013), p. 292	Quantitative	46	31.70	8.95	22–57
Taylor, Fisackerly, Mauren, and Taylor (2013), p. 373	Qualitative	325	20.04	1.22	18–25
Nelson, Delgadillo, and Dew (2013), p. 553	Quantitative	5,097	45.72	14.31	23–96
Schneider and Grimps (2013), p. 612	Quantitative	5,213	56.60	4.43	40–64
		6,391	55.60	5.12	40–64
Silliman and Schumm (2013), p. 652	Quantitative	614	11.67	1.04	10–13
		486	15.19	1.07	14–17
Hughes (2013), p. 700	Quantitative	362	47.00	9.91	24–74
			45.00	9.37	23–65
Gana, Saada, and Untas (2013), p. 759	Quantitative	146	52.97	18.62	18–78
			51.17	18.66	18–83
Hertlein, Emmers-Sommer, and Kennedy (2014), p. 116	Quantitative	871	21.25	5.05	17–60
Chaney (2014), pp. 134–135	Qualitative	30	35.71	7.23	21–45
		30	37.32	6.81	21–45
		30	28.27	1.88	21–45
		30	29.93	7.98	21–45
Burrell, Webb, and White (2014), p. 208	Quantitative	1,348	45.28	17.35	18–89
Rappleyea, Taylor, and Fang (2014), p. 275	Quantitative	1,003	20.70	1.48	18–25
Panetta, Somers, Ceresnie, Hillman, and Partridge (2014), p. 347	Quantitative	195	14.14	1.31	12–18
Cohen and Finzi-Dottan (2014), p. 399	Quantitative	78	39.64	8.20	21–68
Monk, Vennum, Ogolsky, and Fincham (2014), pp. 421–422	Quantitative	246	19.50	1.40	18–29
Yahya and Boag (2014), p. 488	Qualitative	57	21.90	8.80	18–62
Dhar (2014), p. 539	Qualitative	15	22.40	2.80	19–27
Chaney et al. (2014), pp. 566, 568	Quantitative	60	29.10	7.32	20–45
Tang, Curran, and Arroyo (2014), p. 605	Quantitative	280	23.20	5.70	18–58
Jensen, Shafer, and Larson (2014), p. 645	Quantitative	2,030	35.97	10.25	18–77
		488	43.92	10.41	18–77
Sanders (2012), p. 27	Quantitative	1,436	25.79	4.45	18–34
Carlson et al. (2012), p. 85	Quantitative	393	34.18	7.96	19–70
		393	31.80	6.68	20–56
Chaney (2012), p. 196	Qualitative	17	40.06	1.23	23–61
Dijkstra, Barelds, Groothof, Ronner, and Nauta (2012), p. 100	Quantitative	354	40.18	10.44	17–71
		558	43.57	12.52	18–72
Antonopoulou, Alexopoulos, and Maridaki-Kassotaki (2012), p. 298	Quantitative	190	11.26	0.73	9.42–13.83
Mortensen and Barnett (2015), p. 6	Quantitative	82	2.31	0.30	1.83–2.92
Emmers-Sommer (2015), p. 269	Quantitative	717	22.23	5.53	18–63
Horvath and Lee (2015), p. 342	Quantitative	166	15.50	1.13	14–17
Lesch and Scheffler (2015), p. 467	Qualitative	14 (12)	15.42	0.79	14–17
		14 (12)	44.50	3.73	39–51
Xia et al. (2015), p. 496	Quantitative	589	15.76	1.33	12–19
Hall (2015), p. 719	Quantitative	1,490	21.14	2.14	18–30

Note. The standard deviations of 1.88 (Chaney, 2014) and 1.23 (Chaney, 2012) were deemed errors because it was not possible to obtain that low of standard deviations using at least one value of the lowest range and one value of the highest range and using most other numbers close to the mean score (which will yield the lowest possible standard deviation). For Chaney (2014) a standard deviation of 3.67 was the lowest obtained and for Chaney (2012) a standard deviation of 6.75 was the lowest obtained.

Table 2. Descriptive data for number of children or number of siblings for respondent compared with range divided by four estimates for standard deviations from five articles published in Marriage & Family Review, 2012–2015.

Study authors, date of publication, page number(s) containing data	Type of study	N	Mean	SD	Range
Squire et al. (2013), p. 139	Quantitative	356	2.45	1.67	1–10
Thompson et al. (2013), p. 153	Qualitative	20	2.20	0.92	1–3
Higginbotham and Agee (2013), p. 181	Quantitative	447	.93	1.42	0–9
			.93	1.46	0–10
Sanders (2012), p. 27	Quantitative	1,436	2.46	1.93	0–10
Carlson et al. (2012), p. 85	Quantitative	393	2.37	1.39	0–11
		393	2.38	1.39	0–11

Table 3. Descriptive data for duration of relationship (years) for respondents compared with range divided by four estimates for standard deviations from four articles published in Marriage & Family Review, 2012–2015.

Study authors, date of publication, page number(s) containing data	Type of study	N	Mean	SD	Range
Thompson et al. (2013), p. 153	Qualitative	20	15.20	5.35	7–22
Hughes (2013), p. 700	Quantitative	355	16.63	11.32	.5–42
Gana et al. (2013), p. 759	Quantitative	146	28.33	17.30	.33–60
Tang et al. (2014), p. 605	Quantitative	280	2.42	20.17	.17–10.0

Table 4. Descriptive data for education of respondents compared with range divided by four estimates for standard deviations from seven articles published in Marriage & Family Review, 2012–2015.

Study authors, date of publication, page number(s) containing data	Type of study	N	Mean	SD	Range
Higginbotham and Agee (2013), p. 181	Quantitative	447	14.42	1.98	9–17
			14.37	1.99	9–17
Nelson et al. (2013), p. 553	Quantitative	5,097	12.95	2.84	0–20
Silliman and Schumm (2013), p. 652	Quantitative	614	6.86	.86	5–8
		486	10.27	1.17	7–12
Burrell et al. (2014), p. 208	Quantitative	1,348	13.31	2.81	0–20
Chaney et al. (2014), pp. 566, 568	Quantitative	60	12.03	1.57	8–17
Sanders (2012), p. 27	Quantitative	1,436	13.46	1.90	7–20
Xia et al. (2015), p. 496	Quantitative	589	9.67	1.02	7–12

Analysis procedures

For each type of variable, the reported range was divided by the reported standard deviation to create a raw data ratio (rawratio) for each set of data. The average of those ratios was compared with ratios of 4 or 5 or 6, using one-sample t-tests. The ratios were computed for each of the four demographic variables and for the total data set. It might not be intuitively obvious why the ratios obtained by dividing the range by actual standard deviations would be useful.

Brase and Brase (2015) recommend the formula SDest = range/4. Algebra allows us to rearrange the formula to Range/SDest = 4. If the formula is correct, then dividing the actual range by the actual standard deviation [Range/SDact] from Tables 1 through 4 should yield values close to 4. If the values are greater or less than 4, then that would suggest that the original formula was not working very well. We used the recommendations of Hozo et al. (2005) and Wan et al. (2014) for additional comparisons.

Results

Observed ratios of range to standard deviation for age, duration of relationship, number of children or siblings, and years of education

For age of respondents, 47 sets of data from 35 articles were used. The average rawratio (range divided by standard deviation) value was 4.71 (SD = 1.44) with a median of 4.58 (range, 2.65–8.51). For duration of relationship, four sets of data from four articles were used; the average rawratio value was 3.94 (SD = 1.32) with a median of 3.56 (range, 2.80–5.85). For number of children or siblings, seven sets of data from five articles were used; the average rawratio value was 5.97 (SD = 1.99) with a median of 6.34 (range, 2.17–7.91). For years of education, nine sets of data were used from seven articles; the average rawratio value was 5.27 (SD = 1.44) with a median of 4.90 (range, 3.49–7.12). For all 67 points of data, the average rawratio value was 4.87 (SD = 1.54) with a median of 4.72 (range, 2.17–8.51).

Estimated ratios of range to standard deviation for all variables together

Numerical rules of thumb for estimations

Using a one-sample t-test to compare the overall rawratio of 4.87 with a comparison value of 4.0 yielded t(66) = 4.64, which was statistically significant (p < .001), indicating that using a value of four for the divisor was, on average, an underestimate. Changing the test value to 5.0 yielded a t(66) = −.69, which was not significant, indicating that using a value of five for the divisor was more accurate than using a value of four. However, comparing the average rawratio score to a test value of six using a one-sample t-test yielded a significant t(66) = −6.02 (p < .001), indicating that using six as a divisor into the range would lead to an underestimate of the actual standard deviations.

Using Hozo et al.’s method of estimation

Using Hozo et al.’s (2005) method yielded an average ratio of 5.50 (SD = 0.91), which compared with the rawratio average of 4.87 yielded a significant t(66) = 3.70 (p < .001). Thus, the Hozo et al. (2005) approach appeared to underestimate the actual standard deviations. Redoing the same paired samples t-test for each of the three levels in Hozo et al.’ s approach, yielded nonsignificant results for the first two levels. For our three samples under 16 cases, Hozo et al. would have predicted a ratio of 3.46, whereas the actual rawratio was 3.29 (SD = 0.47); for our 13 samples between 16 and 70 cases, the ratios were 4 and 3.73 (SD = 0.94), respectively. For our 51 samples over 70 cases, the rawratio was 5.25 (SD = 1.50) compared with Hozo et al.’s estimate of 6.0, yielding t(50) = 3.55 (p = .001). At all three levels, the Hozo et al. (2005) method would have underestimated the standard deviations; however, for samples of 70 or less, the Hozo et al. approach worked reasonably well. For larger samples than 70 cases, dividing the range by five worked better than using the Hozo et al. divisor of six.

Using Wan et al.’s method of estimation

Wan et al. (2014) suggested using a divisor into the range created by using twice the inverse function of z = (n − .375/n + .25). For example, for n = 486, z = .99871. Using the qnorm(z) routine in the R statistical package (Braun & Murdoch, 2007; Dalgaard, 2008), the inverse function of z = 3.0138, which multiplied by two yields an estimated divisor of 6.03. For sample sizes of 50 or below, Wan et al. (2014, p. 4) provided a table of divisors; for samples of over 50, the estimated divisors were calculated for each sample size in our data. We labeled Wan et al.’s divisor estimates as wanratio.

Using Wan et al.’s (2014) method yielded an average wanratio of 5.57 (SD =1.03), which compared with the rawratio average of 4.87, with a significant t(66) = 4.19 (p < .001). Thus, the Wan et al. (2014) approach appeared to underestimate the actual standard deviations by approximately the same degree as the Hozo et al. (2005) approach. Redoing the same paired samples t-test for each of the three levels in Hozo et al.’s approach yielded a nonsignificant result for the first level with a wan ratio of 3.33 (SD = 0.12) very close to the rawratio of 3.29 (SD = 0.47); for our 13 samples between 16 and 70 cases, the wanratio and rawratio were 4.22 (SD = 0.35) and 3.73 (SD = 0.94), respectively, yielding t(12) = 2.49 (p < .05). For our 51 samples over 70 cases, the rawratio was 5.25 (SD = 1.50) compared with a wanratio of 6.05 (SD = 0.59), yielding t(50) = 3.74 (p < .001). At all three levels, the Wan et al. (2014) method would have underestimated the standard deviations; however, for samples of 15 or less, the Wan et al. approach worked reasonably well. For larger samples than 70 cases, dividing the range by 5 worked better than using the Wan et al. divisor of 5.57.

Because Wan et al. (2014) provided a table for divisor values up to n = 50, we reran the analyses for our 12 cases where n < 50. The rawratio was 3.33 (SD = 0.58) compared with a Hozo ratio of 3.86 (SD = 0.25) and a Wan ratio of 3.87 (SD = 0.41), the latter both significantly different from the rawratio by paired samples t-tests. Both of the latter approaches overestimated the actual ratio of range to standard deviation.

Results for demographic variables separately

How did the results vary for the different variables? Selecting five as the divisor into the range led to nonsignificant differences from rawratio for all four variables, whereas selecting four as the divisor led to significant differences from one-sample t-tests for three of the four variables (excluding duration of relationship). Selecting six as the divisor led to a significant difference only for age (p < .001) and nearly so for duration of relationship (p < .06). Overall, if one wanted to pick a number as a divisor into the range for estimating the standard deviation, five appeared to work best, although not as well as four in the case of duration of relationship. Furthermore, selecting five worked better than using either the Hozo ratio or the Wan ratio for age, duration of relationship, and years of education, whereas the latter two approaches worked better, equally well, than selecting five.

Follow-up analyses

While studying the raw data, it seemed to us that when the mean was relatively closer to the lower or higher levels of the range (as opposed to being centered in the middle of the minimum and maximum values) that the prediction of the standard deviation was less accurate. To test this idea, we created an indirect measure of skew (labeled SKEWNEW) = [mean – minimum]/[maximum – minimum]. If the mean was the same as the minimum value, then SKEWNEW = 0. If the mean was centered between the minimum and maximum, then SKEWNEW = .5. If the mean was the same as the maximum, then SKEWNEW = 1.0. Figure 1 shows the plot of SKEWNEW versus ratio where ratio = [maximum – minimum]/SD. The quadratic curve has an R² of .392, which is larger than the linear curve’s R² of .261. To try to linearize the pattern, we calculated the absolute value of the difference between SKEWNEW and .50 (essentially calculating a deviation of the mean from the center of the range), so that a SKEWNEW of .00 or 1.00 would have a new value of .50 (recoded SKEWNEW = SKEWMAG for magnitude of SKEWNEW). Having done this yielded Figure 2 where the quadratic pattern’s R² was not much greater at .393 than the R² (.377) for the linear model in which RAWRATIO = 3.56 + 8.40(SKEWMAG).

Figure 1. Ratio (range divided by standard deviation) as a function of skew ([mean – minimum] divided by [maximum – minimum]).

Figure 2. Ratio (range divided by standard deviation) as a function of SKEWMAG (absolute value of the deviation of the mean from its center of range).

Revised formula based on skew

That formula suggested a revised ratio pattern, labeled SKEWRATIO as follows, depending on SKEWMAG. If SKEWMAG is .00 to .02, divide the range by 3.50; if .03 to .04, divide the range by 4; if .05 to .11, divide by 4.5; if.12 to .17, divide by 5.0; if .18 to .23, divide by 5.5; if .24 to .29, divide by 6.0; if .30 to .35, divide by 6.5; if .36 to .41, divide by 7.0; if .42 to .47, divide by 7.5; and if .47 to .50, divide by 8.0. The overall difference between SKEWRATIO (M = 5.06, SD = 0.96) and RAWRATIO was not significant and the closeness of SKEWRATIO to five suggests that using five as a divisor into the range to estimate the standard deviation is more accurate than using a divisor of four or six. However, using Hozo et al.’s three levels of sample size, SKEWRATIO provided less accurate estimates of rawratio than did any other approach for samples less than n = 70. For samples of 70 or greater, SKEWRATIO had a mean of 5.12 (SD = 1.02), which was not significantly different from rawratio. Predicting rawratio from sample size and SKEWMAG yielded the equation 3.43 + .00021(SAMPLESIZE) + 8.26(SKEWMAG); however, the effect of sample size was not significant (p < .08), whereas the effect of SKEWMAG remained significant (p < .001). Even though sample size was not a significant predictor, there was still an apparent pattern in which RAWRATIO did not ever exceed four unless the sample was 49 or greater, did not ever exceed six unless the sample was 170 or greater, and did not ever exceed eight unless the sample was greater than 870. On the other hand, RAWRATIO did not ever drop below three until sample size was below 615 and did not ever drop below four unless the sample size was less than 1,450. Thus, sample size seems to constrain the possibilities for the ratio of range divided by standard deviation even if sample size per se does not strongly predict that ratio.

How to get the best estimates for standard deviations

No single approach worked for all sample sizes. For small samples (n ≤ 15), the use of Table 1 in Wan et al. (2014) provided the best estimates here. For samples between 16 and 70, dividing by four (Hozo et al.’s approach) seemed to work better. For samples larger than 71, the SKEWRATIO approach seemed to work better, although simply using five as a divisor into the range would work nearly as well in addition to being far less complex to calculate. In terms of the four separate demographic variables, it was notable that the SKEWRATIO method worked as well or better than using five as a divisor, or using the Hozo or Wan ratios. For age, the skew ratio was 5.00, closer than any other method (Hozo, 5.41; Wan, 5.47) to the rawratio of 4.71 and equal to dividing by five; for duration of relationship, the skew ratio was 4.75, closer than any method (Hozo, 5.50; Wan, 5.13) to the rawratio of 3.94 except for dividing by four; for number of children or siblings, the skew ratio was 6.14, closer than any method (Hozo, 5.71; Wan, 5.71) to the rawratio of 5.97 except for dividing by six; and the skew ratio for education was 4.72, closer than any method (Hozo, 5.78; Wan, 6.17) to the rawratio of 5.27 except for dividing by five, which was about the same error in the other direction.

Discussion

Future research might focus on larger studies or studies from other scholarly sources, including journals in more related (e.g., Journal of Marriage and Family) or less related (e.g., Journal of Abnormal Psychology) fields. Future research should extend the analysis from demographic variables to social-psychological measures, both in terms of single items and multi-item scales.

Overall, our results suggest that, ceteris paribus, it would be better to revise Brase and Brase’s rule to divide the range by five rather than to divide by four. However, as we demonstrated with this particular set of data, the magnitude of the relative position of the mean with respect to the total range of the variable changed the best divisor to use, from 3.5 to as high as 8. Determining the generalizability of this finding to other data sets is beyond the scope of this report but could be a useful direction for future research. Our results might not work as well for binary variables. Because our skew ratio approach was empirically derived, it might not replicate in a different set of data. More detailed statistical work might lead to a formula that better predicts standard deviations as a function of skew as defined here in terms of means or otherwise in terms of the median.