# What Does It “Mean”? A Review of Interpreting and Calculating Different Types of Means and Standard Deviations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1/2}) for a given drug product). The study design is not intended for a comparison across effects (e.g., no inter-treatment comparison).

## 2. Estimation of Means and Variances

#### 2.1. Illustrating the Problem

#### 2.2. Within Study Estimation—Ungrouped Data

#### 2.2.1. Arithmetic Mean and Standard Deviation

_{i}, by using the index, i, to indicate each integer from 1 to n, in turn.

#### 2.2.2. Harmonic Mean and Standard Deviation

#### Harmonic Mean

_{1/2}values is always greater than the HM of the individual T

_{1/2}values, there is no difference in the estimate of the T

_{1/2}mean when estimated as 0.693/(arithmetic mean of λ

_{z}) or as the HM of T

_{1/2}values. The reason for this relationship is detailed in Appendix C.

#### Estimation of the Harmonic Standard Deviation

_{i}is the original observation and n is the number of observations.

_{1/2}is computed on the basis of the reciprocal λ

_{z}values, to return to an estimate of the stdev associated with the T

_{1/2}, it is necessary to divide by the constant 0.693. Indeed, it is this estimate of the stdev which should be used when describing the variation of individual values about a HM. The use of stdev values estimated on the basis of the arithmetic T

_{1/2}is appropriate only when used to describe the variability about an arithmetic mean. Such values should not be reported when the T

_{1/2}is summarized as the HM.

#### 2.2.3. Geometric Means and Standard Deviations

#### Estimation of the Geometric Mean

_{max}). It is also used for the estimation of means when the values are multiplied rather than added together. Both of these situations are described below.

#### Multiplicative Relationships

^{th}root of the product (denoted by the symbol $\mathsf{\Pi}$) of the n positive observations.

^{τ/λ}

- Δ = the kind of change (where Δ = 2 for doubling; Δ = 0.5 for halving)
- λ = the time associated with a doubling or halving of bacterial number
- τ = the duration of time
- Y = the resulting bacterial number

^{20/10}= 400

^{120/35}= 37.15

^{60/10}= 2377.6

^{τ/λ}= (0.5)

^{10/35}= 0.820

^{2}× 0.820

^{12}× 2

^{6})

^{1/20}= 1.172. Table 8 shows the calculation of the expected number of bacteria at the end of each 10-min interval and the calculation of the geometric mean of the 10-min interval rates. In addition, the table shows that replacing each 10-min interval rate with the geometric mean of the 10-min interval rates results in the same calculation of number of bacteria at the end of 200 min.

_{0–200}~ 57982, the mean CFU/min ~ 290.

#### Ln-Transformed Data

_{max}. Ln-transformation of these pharmacokinetic parameter values is based upon an assumption that they are better described in terms of a log-normal rather than a normal distribution. Because Ln(a) + Ln(b) = Ln(ab), the process of adding Ln-transformed values is in fact a multiplicative computation. Furthermore, because Ln (ab)

^{1/n}= (

^{1}/n)Ln(ab) and exp[Ln (ab)

^{1/n}] = (ab)

^{1/n}, exponentiation of the averaged Ln-transformed values reverses the estimation procedure from one of addition (Ln-transformed values) to that of multiplication when expressed on the original scale.

_{1/2}, these reflect C

_{max}values (Table 9). The difference between the arithmetic mean of the untransformed data versus the geometric mean (based upon the Ln-transformed dataset), is provided below. Again, the arithmetic mean of the Ln-transformed data are exponentiated to obtain the geometric mean (i.e., 10.4 = exp

^{2.34}).

#### Estimation of the geometric standard deviation (for Ln-transformed values)

_{i}are the individual observations, with i ranging from 1 to n [4].

#### 2.3. Least Square (Marginal) Means Grouped Data

#### 2.3.1. Estimation of the Mean

#### 2.3.2. Estimation of the Stdev about the LSmean

_{max}. Nevertheless, this exercise underscores why, from a statistical perspective, calculation of the 90% confidence interval could be biased in the face of statistically significant sequence effects.

#### 2.4. Cross Study Comparisons

#### 2.4.1. Estimating the Population Mean

_{i}(i.e., the number of subjects included in study i), a sample average ${\overline{X}}_{i}$, and a standard deviation stdev

_{i}.

^{1}/

_{8}

^{th}(

^{1}/

_{4}×

^{1}/

_{2}) of the overall influence on means and variances while each subject in Study 2 would contribute only

^{1}/

_{40}

^{th}(

^{1}/

_{20}×

^{1}/

_{2}) of the overall influence. We need to adjust the values so that each subject in each study contributes only

^{1}/

_{24}

^{th}to the estimate of the population mean.

**×**6.2) + (5

**×**5.5) + (8

**×**6.1) + (12

**×**6.8)]/35

#### 2.4.2. Estimating the Population Variance

_{i}estimates the (square root of) variation within the study from which it was calculated. Just as with the samples averages, one might think of estimating the population within study stdev as the average of the sample stdevs, deriving an average within-study standard deviation (SD

_{WA}) as follows.

_{BA}) as the stdev among the study means as follows:

_{WA}or SD

_{BA}will seriously underestimate the total population variance. The most straightforward way to think about calculating an estimate of the population variance from the estimates of within and between study stdevs would be to square the stdevs and to add them.

_{w}is the weighted sum of the individual study variance estimates.

_{B}, is the weighted sum of the squared deviations of the individual study means from the LSmean.

## 3. Closing Comments

## Conflicts of Interest

## Appendix A

_{1}and n

_{2}, with marginal means ${\overline{\mathrm{X}}}_{1}$ and ${\overline{\mathrm{X}}}_{2}$:

## Appendix B

^{2}− 2AB + B

^{2}≥ 0.

## Appendix C

#### Relationship of Terminal Elimination Rate Constant and the T_{1/2}

_{z (}λ

_{z}) is the elimination rate constant at the z

^{th}phase of the decay portion of the profile, estimated as the slope of the line. For the sake of this discussion, assume that the drug in question follows a one-compartment open body model and therefore “z” is the terminal phase of decline.

_{z}× (t),

_{z}× (t),

_{z}× (T

_{1/2}).

_{z}, we can solve for T

_{1/2}:

_{z}) = T

_{1/2}

_{z}= T

_{1/2}= 0.693/λ

_{z}

**Figure A1.**Depletion curve shown on the linear scale. The curve is best described by the relationship: Y = 100 × exp

^{−0.1735x}; where X = time (h).

**Figure A2.**Depletion curve shown for Ln-transformed values. The line is best described by the equation Ln(Y) = −0.1735X + 4.6052, where X = time (h).

**Table A1.**Comparison of arithmetic vs. HM for T

_{1/2}estimates (including both the values and the codes for estimating these values based upon the columns and rows that might be found in an Excel spreadsheet).

Data Row | Column C | Column D | Column E | |||
---|---|---|---|---|---|---|

T_{1/2} | λ_{z} | 1/T_{1/2} | T_{1/2} | 0.693/λ_{z} | T_{1/2} | |

Row 2 | 11 | 0.063 | 0.091 | |||

Row 3 | 7 | 0.099 | 0.143 | |||

Row 4 | 9 | 0.077 | 0.111 | |||

Row 5 | 4 | 0.173 | 0.25 | |||

Row 6 | 10 | 0.069 | 0.1 | |||

Row 7 | 12 | 0.058 | 0.083 | |||

Row 8 | 23 | 0.03 | 0.043 | |||

Row 9 | 15 | 0.046 | 0.067 | |||

Row 10 | 7 | 0.099 | 0.143 | |||

Row 11 | 18 | 0.039 | 0.056 | |||

Arithmetic mean | 11.6 | 0.075 | sum(C2:C11)/10 | |||

0.693/(Arith mean λ_{z}) | 9.202 | 0.693/(sum(D2:D11)/10) | ||||

Harmonic mean of T_{1/2} | 9.202 | 1/(sum(E2:E11)/10) |

## Appendix D

#### Cross-Study Variability Estimation Procedure (Background)

_{ij}is the jth observation from group I, basic textbook introductions [1] to analysis of variance show that the SS

_{T}, the total sum of squared deviations from the grand mean, ${\overline{X}}_{..}$, can be divided into two sums of squares, as illustrated below, because their cross products sum to zero.

_{W}. As discussed prior to Equation (21), by definition of the sample Stdev for study i, the contribution of study i to SS

_{W}is equal to (n

_{i}− 1) ${S}_{i}^{2}$ and the SS

_{W}is the sum of the contributions from all studies.

_{T}is the SS

_{B}. It can be simplified as displayed in Equation (22) because there are no terms involving the index j.

_{W}is estimated by (N − I)${{\overline{S}}_{.}}^{2}$

_{B}with equal n.

_{W}and SS

_{B}are put back over N − 1 to estimate the total variance, the weights given to each component are seen more clearly.

## References

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**Figure 3.**The distribution of weight values for the no exercise group predicted using the LSmean and a stdev estimate based on the standard error (SE) of the LSmean when assuming that the sample was generated from a single normal population.

**Figure 4.**Distribution of female weights based upon the mean and variance of the sample values of body weights of females in the no exercise group. The X-axis represents body weight (lbs) and the Y-axis represents the fraction of the population of females at that body weight based upon the information contained within subset of individuals included in this hypothetical study. The possible values of the Y-axis range from a value of zero (no individuals expected at that body weight) to 1 (all individuals in the population will have the identical body weight).

**Figure 5.**Distribution of male weights based upon the mean and variance of the sample values of body weights of the males in the no exercise group.

**Figure 6.**Distribution of population of male and female body weights based upon the mean and variance of the sample values from the no-exercise group, assuming an equal likelihood of sampling from either gender.

**Table 1.**Comparison of statistics based on original values and Ln-transformed values back-transformed to the original units. Stdev: standard deviation.

Transformation Used | Mean | Mean ± Stdev | Mean ± 2 Stdev | Mean ± 3 Stdev |
---|---|---|---|---|

Untransformed | 11.6 | 5.7, 17.3 | 0.2, 23 | −5.5, 28.7 |

Back-transformed from Ln | 10.4 | 6.2, 17.3 | 3.7, 28.8 | 2.2, 47.9 |

**Table 2.**Example of differing summary values depending upon the underlying assumptions and estimation procedures for obtaining the mean.

Number | Ln number | 1/Number | |

11 | 2.4 | 0.09 | |

7 | 1.95 | 0.14 | |

9 | 2.20 | 0.11 | |

4 | 1.39 | 0.25 | |

10 | 2.30 | 0.10 | |

12 | 2.48 | 0.08 | |

23 | 3.14 | 0.04 | |

15 | 2.71 | 0.07 | |

7 | 1.95 | 0.14 | |

18 | 2.89 | 0.06 | |

Type of estimate | Arithmetic | Geometric | Harmonic |

Mean | 11.60 | 10.38 | 9.20 |

Stdev | 5.70 | 5.29 | 5.06 |

%CV | 49.14 | 50.96 | 50.03 |

**Table 3.**Matching the selection of mean to the nature of the distribution and the question being addressed.

Type of mean and standard deviation | Nature of distribution (examples) | Considerations for its application |
---|---|---|

Arithmetic | Normal (i.e., additive) | When the estimate of interest is based upon the sum of the individual observations and the data are best presented as a normal distribution. |

Harmonic | Reciprocal transformation of positive real values | In general, the harmonic mean (HM) is useful for expressing average rates (e.g., miles per hour; widgets per day). In clinical pharmacology, an elimination rate constant, λ_{z}, is estimated for each subject based on his/her systemic drug concentration during the depletion portion of the Ln-concentration vs. time profile. The corresponding estimate of the elimination half-life (T_{1/2}) is then derived on the basis of the elimination rate constant (see Appendix C for more details on calculating the terminal elimination rate where we show that T_{1/2} = Ln 2/λ_{z} or 0.693/λ_{z}). Because of the relationship of T_{1/2} to the reciprocal value of λ_{z} (the latter being a rate constant), harmonic means may be used when describing the average time to reduce the systemic drug concentrations by 1/2. It should be noted that if the mean T_{1/2} was generated by obtaining the arithmetic average of the T_{1/2} values, then that estimate should be referred to as an arithmetic mean and not as a HM. The mean T_{1/2} represents a HM only when the actual averaging was done on the basis of λ_{z} estimates. It is only in that case (i.e., when the mean T_{1/2} was derived via transformation of the HM of λ_{z}) that we have the HM of T_{1/2}. |

Geometric | Log transformation of positive real values | Within the realm of pharmacokinetics, geometric means are typically used when describing the means of variables such as area under the curve (AUC) and maximum concentrations (C_{max}). These variables are often transformed to the natural logarithm (Ln) prior to analysis and the geometric mean computed using the back-transformation shown in Equation (11). This type of assessment is also important for estimating the average performance of an investment where the interest rates are compounded over time or when averaging changes in bacterial growth rates. |

Least square means | Should be consistent with the distribution characteristics of the data collected and the model used to address the study assumptions and investigation | The use of least square means is important when there are an unequal number of observations associated with any of the terms in the statistical model. |

**Table 4.**Comparison of arithmetic vs. harmonic standard deviation (Stdev) values and Excel codes for T

_{1/2}.

Data Row | Column C | Column D | Column E | Column F | |
---|---|---|---|---|---|

T_{1/2} | λ_{z} | T_{1/2} sq′d Deviations | λ_{z} sq′d Deviations | ||

Row 2 | 11 | 0.063 | (C2–C12)^{2} | (D2–D12)^{2} | |

Row 3 | 7 | 0.099 | (C3–C12)^{2} | (D3–D12)^{2} | |

Row 4 | 9 | 0.077 | (C4–C12)^{2} | (D4–D12)^{2} | |

Row 5 | 4 | 0.17325 | (C5–C12)^{2} | (D5–D12)^{2} | |

Row 6 | 10 | 0.0693 | (C6–C12)^{2} | (D6–D12)^{2} | |

Row 7 | 12 | 0.05775 | (C7–C12)^{2} | (D7–D12)^{2} | |

Row 8 | 23 | 0.05775 | (C8–C12)^{2} | (D8–D12)^{2} | |

Row 9 | 15 | 0.0462 | (C9–C12)^{2} | (D9–D12)^{2} | |

Row 10 | 7 | 0.099 | (C10–C12)^{2} | (D10–D12)^{2} | |

Row 11 | 18 | 0.077 | (C11–C12)^{2} | (D11–D12)^{2} | |

Row 12 | Arithmetic mean | 11.6 | 0.0753 | ||

Row 13 | Harmonic mean of T_{1/2} | 9.20 | |||

Arithmetic Stdev | 5.70 | sqrt[(sum(E2:E11))/9] | |||

Harmonic Stdev | 5.06 | [D13^{2} × sqrt[(sum(F2:F11))/9]]/0.693 |

**Table 5.**Comparison of estimating average change when expressed as an arithmetic versus as a geometric mean when there are compounded (multiplicative) changes.

Data Row | Interest (percent) | Column D relative change | Excel code for geometric mean | |

Row 2 | 0.25 | 1.25 | - | |

Row 3 | −0.36 | 0.64 | ||

Row 4 | 0.18 | 1.18 | ||

Row 5 | 0.14 | 1.14 | ||

Row 6 | −0.75 | 0.25 | ||

Arithmetic mean | - | −0.108 | - | |

Geometric mean | 0.769 | (D2 × D3 × D4 × D5 × D6)^{(1/5)} | ||

Proportion remaining | - | 0.892 | - | - |

Doubling time or halving time (λ) | Time duration (τ) | Value of “X” | Relative change (Δ^{τ/λ}) | Resulting # bacteria (Y) |
---|---|---|---|---|

10 | 20 | 100 | 4 | 400 |

35 | 120 | 400 | 0.0929 | 37.150 |

10 | 60 | 37.15 | 64 | 2377.591 |

Data Row | Column A | Column B | Column C | Column D | Column E |
---|---|---|---|---|---|

Doubling time or halving time | Time duration | Value of “X” | Relative change | Resulting # bacteria (Y) | |

Row 1 | 10 | 20 | 100 | 2^{B1/A1} | C1 × D1 |

Row 2 | 35 | 120 | 400 | 0.5^{B2/A2} | C2 × D2 |

Row 3 | 10 | 60 | 37.15 | 2^{B3/A3} | C3 × D3 |

Column A | Column B | Column C | Column D | Column E | Column F | Column G |
---|---|---|---|---|---|---|

Row | Minutes | 10 min change rate change (∆) | Number of bacteria (Y) based on 10-min change | Estimating Y [X × ∆^{τ/λ} where τ = 10 and λ = 10] | Number of bacteria based on geometric mean | X × GM (τ/λ) |

2 | 0 | 100 | ||||

3 | 10 | 2 | 200 | D2 × C3 | 117.17 | D2 × $C$26 |

4 | 20 | 2 | 400 | D3 × C4 | 137.28 | F3 × $C$26 |

5 | 30 | 0.82 | 328.13 | D4 × C5 | 160.85 | F4 × $C$26 |

6 | 40 | 0.82 | 269.18 | D5 × C6 | 188.46 | F5 × $C$26 |

7 | 50 | 0.82 | 220.82 | D6 × C7 | 220.82 | F6 × $C$26 |

8 | 55 | 200.00 | D7 × (0.82^{0.5}) | |||

9 | 60 | 0.82 | 181.14 | D8 × (C9^{0.5}) | 258.73 | F7 × $C$26 |

10 | 70 | 0.82 | 148.60 | D9 × C10 | 303.14 | F9 × $C$26 |

11 | 80 | 0.82 | 121.90 | D10 × C11 | 355.18 | F10 × $C$26 |

12 | 90 | 0.82 | 100.00 | D11 × C12 | 416.16 | F11 × $C$26 |

13 | 100 | 0.82 | 82.03 | D12 × C13 | 487.60 | F12 × $C$26 |

14 | 110 | 0.82 | 67.30 | D13 × C14 | 571.31 | F13 × $C$26 |

15 | 120 | 0.82 | 55.20 | D14 × C15 | 669.39 | F14 × $C$26 |

16 | 125 | 50.00 | D15 × (0.82^{0.5}) | |||

17 | 130 | 0.82 | 45.29 | D16 × (C17^{0.5}) | 784.31 | F15 × $C$26 |

18 | 140 | 0.82 | 37.15 | D17 × C18 | 918.96 | F17 × $C$26 |

19 | 150 | 2 | 74.30 | D18 × C19 | 1076.72 | F18 × $C$26 |

20 | 160 | 2 | 148.60 | D19 × C20 | 1261.56 | F19 × $C$26 |

21 | 170 | 2 | 297.20 | D20 × C21 | 1478.14 | F20 × $C$26 |

22 | 180 | 2 | 594.40 | D21 × C22 | 1731.90 | F21 × $C$26 |

23 | 190 | 2 | 1188.80 | D22 × C23 | 2029.22 | F22 × $C$26 |

24 | 200 | 2 | 2377.59 | D23 × C24 | 2377.59 | F23 × $C$26 |

25 | GM | Product (C3:C24)^{(1/20)} | ||||

26 | GM | 1.17 |

Data Row | C_{max} Column D | Ln C_{max} Column E | Ln C_{max} |
---|---|---|---|

Row 2 | 11 | 2.40 | - |

Row 3 | 7 | 1.95 | |

Row 4 | 9 | 2.20 | |

Row 5 | 4 | 1.39 | |

Row 6 | 10 | 2.30 | |

Row 7 | 12 | 2.48 | |

Row 8 | 23 | 3.14 | |

Row 9 | 15 | 2.71 | |

Row 10 | 7 | 1.95 | |

Row 11 | 18 | 2.89 | |

Arithmetic mean | 11.6 | 2.34 | sum(E2:E11)/10 |

Geometric mean (exponentiation of arithmetic mean of Ln values)) | - | 10.4 | exp[sum(E2:E11)/10] |

Column A | Column B | Column C | Column D | ||
---|---|---|---|---|---|

Row | Number | Ln number | Geometric sq′d dev | ||

1 | - | 11 | 2.40 | 0.00 | (B1–D12) |

2 | 7 | 1.95 | 0.15 | (B2–D12)^{2} | |

3 | 9 | 2.20 | 0.02 | (B3–D12)^{2} | |

4 | 4 | 1.39 | 0.91 | (B4–D12)^{2} | |

5 | 10 | 2.30 | 0.00 | (B5–D12)^{2} | |

6 | 12 | 2.48 | 0.02 | (B6–D12)^{2} | |

7 | 23 | 3.14 | 0.63 | (B7–D12)^{2} | |

8 | 15 | 2.71 | 0.14 | (B8–D12)^{2} | |

9 | 7 | 1.95 | 0.15 | (B9–D12)^{2} | |

10 | 18 | 2.89 | 0.30 | (B10–D12)^{2} | |

11 | Sum | - | - | 2.34 | sum(D1:D10) |

12 | Average | - | sum(B1:B10)/10 | ||

13 | Geometric mean | exp(D12) | |||

14 | Arithmetic Stdev | - | |||

14 | Geometric Stdev | 5.29 | D13 × sqrt(D11/9) |

No exercise group | Exercise group | |||

Male | Female | Male | Female | |

210 | 150 | 200 | 138 | |

215 | 168 | 192 | 138 | |

189 | 145 | 176 | 144 | |

196 | 160 | 202 | 154 | |

202 | 166 | 210 | 140 | |

155 | 189 | |||

159 | 176 | |||

149 | 188 | |||

138 | 192 | |||

188 | ||||

Marginal means | 202 | 158 | 192 | 143 |

Arithmetic mean | 173 | 174 | ||

LSmean | 180 | 167 |

**Table 12.**Comparison of stdev based upon an assumption of a single versus bimodal dataset comprising the treatment effect.

LSmean | Description | Within/between calculated Stdev | Simulated Stdev (equal probability of sampling from each group) |
---|---|---|---|

No exercise (T0) | Average of T0 males and T0 females | 25.70 | 25.10 |

Exercise (T1) | Average of T1 males and T1 females | 27.77 | 27.05 |

Males | Average of T0 males and T1 males | 12.57 | 12.95 |

Females | Average of T0 females and T1 females | 13.71 | 13.93 |

Heading | Trainer | # Runners | Average MPH | Stdev | Mean × ni | LSmean |
---|---|---|---|---|---|---|

1 | 10 | 6.2 | 1.24 | 62 | - | |

2 | 5 | 5.5 | 0.55 | 27.5 | - | |

3 | 8 | 6.1 | 0.915 | 48.8 | - | |

4 | 12 | 6.8 | 1.7 | 81.6 | - | |

Sum | - | 35 | - | - | 219.9 | 6.28 |

Data Row | Trainer Col B | # Runners Col C | Average MPH Col D | Stdev Col E | Mean×ni Col F | Lsmean Col G | SS_{W} Col H | SS_{B} Col I | Variance_{WB} Col J | Stdev_{WB} Col K |
---|---|---|---|---|---|---|---|---|---|---|

Row 2 | 1 | 10 | 6.2 | 1.24 | 62 | - | 13.84 | 0.07 | - | - |

Row 3 | 2 | 5 | 5.5 | 0.55 | 27.5 | 1.21 | 3.06 | |||

Row 4 | 3 | 8 | 6.1 | 0.915 | 48.8 | 5.86 | 0.27 | |||

Row 5 | 4 | 12 | 6.8 | 1.7 | 81.6 | 31.79 | 3.21 | |||

Row 6 | - | 35 | - | - | 219.9 | 6.28 | 52.70 | 6.61 | 1.74 | 1.32 |

Data Row | Trainer Col B | # Runners Col C | Ave MPH Col D | Stdev Col E | Mean×ni Col F | Lsmean Col G | SS_{W}Col H | SS_{B}Col I | Variance_{WB}Col J | Stdev_{WB}Col K |
---|---|---|---|---|---|---|---|---|---|---|

Row 2 | 1 | 10 | 6.2 | 1.24 | D2 × C2 | - | (E2^{2}) × (C2-1) | C2 × (D2-$G$6)^{2} | - | |

Row 3 | 2 | 5 | 5.5 | 0.55 | D3 × C3 | (E3^{2}) × (C3-1) | C3 × (D3-$G$6)^{2} | |||

Row 4 | 3 | 8 | 6.1 | 0.915 | D4 × C4 | (E4^{2}) × (C4-1) | C4 × (D4-$G$6)^{2} | |||

Row 5 | 4 | 12 | 6.8 | 1.7 | D5 × C5 | (E5^{2}) × (C5-1) | C5 × (D5-$G$6)^{2} | |||

Row 6 | - | SUM (C2:C5) | - | - | SUM (F2:F5) | F6/C6 | SUM (H2:H5) | SUM (I2:I5) | (I6 + H6)/(C6 − 1) | SQRT(J6) |

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## Share and Cite

**MDPI and ACS Style**

Martinez, M.N.; Bartholomew, M.J.
What Does It “Mean”? A Review of Interpreting and Calculating Different Types of Means and Standard Deviations. *Pharmaceutics* **2017**, *9*, 14.
https://doi.org/10.3390/pharmaceutics9020014

**AMA Style**

Martinez MN, Bartholomew MJ.
What Does It “Mean”? A Review of Interpreting and Calculating Different Types of Means and Standard Deviations. *Pharmaceutics*. 2017; 9(2):14.
https://doi.org/10.3390/pharmaceutics9020014

**Chicago/Turabian Style**

Martinez, Marilyn N., and Mary J. Bartholomew.
2017. "What Does It “Mean”? A Review of Interpreting and Calculating Different Types of Means and Standard Deviations" *Pharmaceutics* 9, no. 2: 14.
https://doi.org/10.3390/pharmaceutics9020014