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Mean & SD conversions

Mean and standard deviation (SD) conversions

In just one click, all the data types below (for one group or more) will be converted to mean and SD, and then will be automatically converted to mean and SD pre- (baseline) and post-treatment to mean and SD change. N.B: The correlation coefficient is calculated if available or assumed if not.

1. Mean and standard deviation (SD) Change

A) Needed values

To calculate the change in mean (x̄) and standard deviation (SD) from baseline, you need the following values:

  1. Mean (x̄) (baseline)

  2. Mean (x̄) post-treatment (Final)

  3. Standard deviation (SD) (Baseline)

  4. Standard deviation (SD) (Final)

  5. Correlation coefficient (C.C)

B) Steps

  1. In the Input table: put each of the previous values into the corresponding cells
  2. Click submit
  3. In the Output table: you will get the mean and SD change

C) Equations

The output was calculated upon the following equations:

1) Mean (x̄) Change:

xˉ Change = xˉ(Final)−xˉ(Baseline)\textbf{x̄ Change = }x̄ (Final) - x̄ (Baseline)

2) Standard deviation (σ) change:

SD Change = SD Baseline2+SD Final2−(2×C.C×SD Baseline×SD Final)\textbf{SD Change = }\sqrt{\mathrm{SD}{\text { Baseline}}^{2}+\mathrm{SD}{\mathrm{} \text { Final}}^{2}-\left(2 \times \mathrm{C.C} \times \mathrm{SD}{\text { Baseline}} \times \mathrm{SD }{\mathrm{ } \text { Final}}\right)}

D) Citation and equations source

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.4 (updated August 2023). Cochrane, 2023. Available from www.training.cochrane.org/handbook (opens in a new tab). [Chapter: 6.5.2.8 Imputing standard deviations for changes from baseline, URL: Chapter 6: Choosing effect measures and computing estimates of effect | Cochrane Training (opens in a new tab)].

E) Additional notes

1) In the meta-analysis, we usually need to calculate C.C if the data presented contains the following:

A) Standard deviation (SD) (Baseline)

B) Standard deviation (SD) (Final)

C) Standard deviation (SD) (Change)

The output (C.C)  will be calculated upon the following equation:

C.C = SD Baseline2+SD Final2−SD Change22×SD Baseline×SD Final \textbf{C.C = }\frac{\mathrm{SD}{\text { Baseline}}^{2}+\mathrm{SD}{\mathrm{} \text { Final}}^{2}-\mathrm{SD}{\mathrm{} \text { Change}}^{2}}{2 \times \mathrm{SD}{\text { Baseline}} \times \mathrm{SD}{\mathrm{} \text { Final }}}

2) A- If C.C is available from more than one study, we use their average.

B- If only one C.C is reported or calculated, we use it.

C- If the previous values (SD Baseline, SD Final, and SD Change) were unavailable in all studies, C.C is assumed to be 0.50. (Abrams, K. R., Gillies, C. L., & Lambert, P. C. (2005). Meta‐analysis of heterogeneously reported trials assessing change from baseline. Statistics in medicine, 24(24), 3823-3844).

2. Mean and confidence interval (CI)

A) Needed values

To estimate the mean and standard deviation (SD) from mean and 95% confidence intervals (CI), you need the following values:

  1. Mean

  2. Upper limit of CI

  3. Lower limit of CI

  4. Sample size (N)

B) Steps

  1. In the Input table: put each of the previous values into the corresponding cells
  2. Click submit
  3. In the Output table: you will get the data presented as mean and standard deviation

C) Equations

The output was calculated upon the following equations:

1) SD:

SD = N×UpperLimit−LowerLimit3.92\textbf{SD = }\sqrt{\mathrm{N}} \times \frac{Upper Limit - Lower Limit }{3.92}

D) Citation and equations source

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.4 (updated August 2023). Cochrane, 2023. Available from www.training.cochrane.org/handbook (opens in a new tab). [Chapter: 6.5.2.2 Obtaining standard deviations from standard errors and confidence intervals for group means, URL: Chapter 6: Choosing effect measures and computing estimates of effect | Cochrane Training (opens in a new tab)].

E) Additional notes

This equation is for 95%CI. For 90% CI, 3.92 should be replaced by 3.29, and for 99% confidence intervals it should be replaced by 5.15.

3. Mean and standard error (SE)

A) Needed values

To estimate the mean and standard deviation (SD) from mean and standard error (SE), you need the following values:

  1. Mean

  2. Standard Error (SE)

  3. Sample size (N)

B) Steps

  1. In the Input table: put each of the previous values into the corresponding cells
  2. Click submit
  3. In the Output table: you will get the data presented as mean and standard deviation

C) Equations

The output was calculated upon the following equations:

1) SD:

SD = SE×N\textbf{SD = }S E \times \sqrt{N}

D) Citation and equations source

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.4 (updated August 2023). Cochrane, 2023. Available from www.training.cochrane.org/handbook (opens in a new tab). [Chapter: 6.5.2.2 Obtaining standard deviations from standard errors and confidence intervals for group means, URL: Chapter 6: Choosing effect measures and computing estimates of effect | Cochrane Training (opens in a new tab)]

E) Additional notes

When performing this conversion, SE for a single group should be used, rather than using the SE of the mean difference between two groups.

4. Median and range

A) Needed values

To estimate mean (x̄) and standard deviation (σ) from median (Md) and range (a-b), you need the following values:

  1. Median (Md)

  2. The smallest value [minimum in the range] (a)

  3. The largest value [maximum in the range] (b)

  4. Sample size (N)

B) Steps

  1. In the Input table: put each of the previous values into the corresponding cells
  2. Click submit
  3. In the Output table: you will get the data presented as mean and standard deviation

C) Equations

The output was calculated upon the following equations:

1) Mean (x̄):

xˉ = ((a+2Md+b)/4)+((a−2Md+b)/4)\textbf{x̄ = } ((a+2Md+b)/4)+((a-2Md+b)/4)

If the sample size was fairly large it can be calculated according to only:

xˉ≃((a+2Md+b)/4)\textbf{x̄}\simeq ((a+2Md+b)/4)

2) Standard deviation (σ):

if(N<25)⟶σ≃R/4≃b−a12if (N\lt25) \longrightarrow \sigma\simeq R/4\simeq \frac{b-a}{\sqrt{12}} if(25<N<60)⟶σ≃R/4≃b−a4if (25\lt N\lt 60)\longrightarrow \sigma\simeq R/4\simeq \frac{b-a}{4} if(N>60)⟶σ≃R/4≃b−a6 if (N\gt60)\longrightarrow \sigma\simeq R/4\simeq \frac{b-a}{6}

D) Citation and equations source

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.4 (updated August 2023). Cochrane, 2023. Available from www.training.cochrane.org/handbook (opens in a new tab). (Chapter: 6.5.2.6 Ranges, URL: Chapter 6: Choosing effect measures and computing estimates of effect | Cochrane Training (opens in a new tab))

Wan X, Wang W, Liu J, Tong T. Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. BMC Med Res Methodol. 2014;14:135. Published 2014 Dec 19. doi:10.1186/1471-2288-14-135

Hozo SP, Djulbegovic B, Hozo I. Estimating the mean and variance from the median, range, and the size of a sample. BMC Med Res Methodol. 2005;5:13. Published 2005 Apr 20. doi:10.1186/1471-2288-5-13

E) Additional notes

Use this conversion with caution, as the data skewness may affect the results.

5. Median and inter quartile range (IQR)

A) Needed values

To estimate mean (x̄) and standard deviation (SD) from median (Md) and inter-quartile range (هضق), you need the following values:

  1. Median (Md)

  2. First Quartile (Q1)

  3. Third Quartile (Q3)

  4. Sample size (N)

B) Steps

  1. In the Input table: put each of the previous values into the corresponding cells
  2. Click submit
  3. In the Output table: you will get the data presented as mean and standard deviation

C) Equations

The output was calculated upon the following equations:

1) Mean (x̄):

xˉ ≃((Q1+Md+Q3)/3)\textbf{x̄ } \simeq ((Q1 + Md + Q3) / 3)

2) Standard deviation (SD):

((Q3 - Q1) / ((2 * z^(-1) ) (0.75n - 0.125 / N+0.25)))

where z is a constant symbol

Approximately (SD):

SD ≃(Q3−Q1/1.35)=(IQR/1.35)\textbf{SD }\simeq (Q3-Q1/1.35) = (IQR/1.35)

D) Citation and equations source

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.4 (updated August 2023). Cochrane, 2023. Available from www.training.cochrane.org/handbook (opens in a new tab). (Chapter: 6.5.2.5 Interquartile ranges, URL: Chapter 6: Choosing effect measures and computing estimates of effect | Cochrane Training (opens in a new tab))

Wan X, Wang W, Liu J, Tong T. Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. BMC Med Res Methodol. 2014;14:135. Published 2014 Dec 19. doi:10.1186/1471-2288-14-135

Hozo SP, Djulbegovic B, Hozo I. Estimating the mean and variance from the median, range, and the size of a sample. BMC Med Res Methodol. 2005;5:13. Published 2005 Apr 20. doi:10.1186/1471-2288-5-13

E) Additional notes

None.

6. P-value of the difference between groups

A) Needed values

To estimate the mean standard deviation (SD) for each group from the mean difference and P-value of the difference between two groups, you need the following values:

  1. P-value (between two groups)

  2. Sample size of Experimental group (Ne)

  3. Sample size of Control group (Nc)

  4. Mean Difference between two groups

B) Steps

  1. In the Input table: put each of the previous values into the corresponding cells
  2. Click submit
  3. In the Output table: you will get the mean and SD (that will be used for both groups)

C) Equations

The output was calculated upon the following equations:

1) P value to t value to SE then to SD

SE = Mean Differencet−value\textbf{SE = }\frac{Mean \:Difference}{t-value}

Then:

SD = SE1NE+1NC\textbf{SD = } \frac{S E}{\sqrt{\frac{1}{N_{E}}+\frac{1}{N_{C}}}}

D) Citation and equations source

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.4 (updated August 2023). Cochrane, 2023. Available from www.training.cochrane.org/handbook (opens in a new tab). (Chapter: 6.5.2.3 Obtaining standard deviations from standard errors, confidence intervals, t statistics and P values for differences in means, URL: Chapter 6: Choosing effect measures and computing estimates of effect | Cochrane Training (opens in a new tab))

Lin, J. T. (1989). Approximating the normal tail probability and its inverse for use on a pocket calculator. Journal of the Royal Statistical Society: Series C (Applied Statistics), 38(1), 69-70.‏

E) Additional notes

If the study provides you with the mean for each group, you might need to compute the mean difference using the formula:

Mean Difference = Mean1−Mean2\textbf{Mean Difference = }Mean1−Mean2

Where Mean 1 is a mean for the first group, Mean 2 is a mean for the second group

When studies report levels of significance, they often use shorthand expressions like "P<0.05" or "P=NS" (which means "not significant" and usually implies P>0.05). Instead of exact P values, a conservative approach is to round up to the nearest limit (e.g., for P<0.05, use P=0.05; for P<0.01, use P=0.01; and for P<0.001, use P=0.001). However, this approach does not work when results are reported as P=NS or P>0.05 .

In a meta-analysis, you will require SDs for both the experimental and comparator groups. These equations compute the average SDs for both arms, allowing you to use this common SD for both groups.

7. Standard error (SE) of the difference between groups

A) Needed values

To estimate the standard deviation (SD) from the standard error (SE) of the difference between two groups, you need the following values:

  1. Standard error between two groups (SE)

  2. Sample size of Experimental group (Ne)

  3. Sample size of Control group (Nc)

B) Steps

  1. In the Input table: put each of the previous values into the corresponding cells
  2. Click submit
  3. In the Output table: you will get the mean and SD (that will be used for both groups)

C) Equations

The output was calculated upon the following equation:

1) SD:

SD = SE1NE+1NC\textbf{SD = }\frac{S E}{\sqrt{\frac{1}{N_{E}}+\frac{1}{N_{C}}}}

D) Citation and equations source

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.4 (updated August 2023). Cochrane, 2023. Available from www.training.cochrane.org/handbook (opens in a new tab). (Chapter: 6.5.2.3 Obtaining standard deviations from standard errors, confidence intervals, t statistics and P values for differences in means, URL: Chapter 6: Choosing effect measures and computing estimates of effect | Cochrane Training (opens in a new tab))

E) Additional notes

In the meta-analysis, you probably need SDs for both the experimental and comparator groups. This equation computes the average SDs for both arms, allowing you to use this common SD for both groups.

8. Confidence interval (CI) of the difference between groups

A) Needed values

To estimate the standard deviation (SD) for both groups from the Confidence interval (CI) of the difference between two groups, you need the following values:

  1. Upper limit of the CI

  2. Lower limit of the CI

  3. Sample size of Experimental group (Ne)

  4. Sample size of Control group (Nc)

  5. CI level (Usually 95% CI)

B) Steps

  1. In the Input table: put each of the previous values into the corresponding cells
  2. Click submit
  3. In the Output table: you will get the mean and SD (that will be used for both groups)

C) Equations

The output was calculated upon the following equations:

1) CI to SE then to SD

SE = UpperLimit−LowerLimit3.92\textbf{SE = } \frac{Upper Limit - Lower Limit }{3.92}

Then:

SD = SE1NE+1NC\textbf{SD = }\frac{S E}{\sqrt{\frac{1}{N_{E}}+\frac{1}{N_{C}}}}

D) Citation and equations source

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.4 (updated August 2023). Cochrane, 2023. Available from www.training.cochrane.org/handbook (opens in a new tab). (Chapter: 6.5.2.3 Obtaining standard deviations from standard errors, confidence intervals, t statistics and P values for differences in means, URL: Chapter 6: Choosing effect measures and computing estimates of effect | Cochrane Training (opens in a new tab))

E) Additional notes

In the meta-analysis, you probably need SDs for both the experimental and comparator groups. These equations compute the average SDs for both arms, allowing you to use this common SD for both groups.

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