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Calculate Population-Representative Standard Deviations

Source: R/w_sd.R
w_sd.Rd

w_sd() calculates standard deviations that accurately represent your population by using survey weights. The standard deviation tells you how spread out your data is around the average – a larger SD means more variation in responses, while a smaller SD means responses cluster tightly around the mean.

Without weights, you describe spread in your sample only. With weights, you estimate how spread out values are in the entire population.

Usage

w_sd(data, ..., weights = NULL, na.rm = TRUE)

Arguments

data

Your survey data (a data frame or tibble)

...

The numeric variables you want to analyze. You can list multiple variables or use helpers like starts_with("trust")

weights

Survey weights to make results representative of your population. Without weights, you get the simple sample standard deviation. Give a column name (unquoted or as a string), an expression such as sampling_weight * 2, or a numeric vector with one weight per row.

na.rm

Remove missing values before calculating? (Default: TRUE). With FALSE, the result for a variable that contains missing values is NA (as in base R).

Value

Population-weighted standard deviation(s) with sample size information, including the weighted SD, effective sample size (effective N), and the number of valid observations used.

Details

Understanding the Results

  • Weighted SD: The population-representative standard deviation. Roughly 68% of your population falls within one SD of the weighted mean.

  • Effective N: How many independent observations your weighted data represents. Always less than or equal to the actual sample size.

  • N / Missing: Valid and missing cases. With weights, both are sums of weights (displayed rounded), as SPSS reports them under WEIGHT BY; Kish's effective N is shown by summary().

A large difference between weighted and unweighted SD suggests that the variability in your sample does not accurately reflect the population.

When to Use This

Use w_sd() when:

  • You need to report how spread out a variable is in the population

  • You want to compare variability across groups with proper weighting

  • Your survey used complex sampling (oversampling, stratification)

  • You need SPSS-compatible weighted standard deviations

Formula

The weighted standard deviation uses the SPSS frequency weights formula:

\(s_w = \sqrt{\frac{\sum w_i (x_i - \bar{x}_w)^2}{V_1 - 1}}\)

where \(V_1 = \sum w_i\) is the sum of all weights and \(\bar{x}_w = \sum w_i x_i / V_1\) is the weighted mean.

The effective sample size is: \(n_{eff} = (\sum w_i)^2 / \sum w_i^2\)

References

IBM Corp. (2023). IBM SPSS Statistics 29 Algorithms. IBM Corporation.

See also

sd for the base R standard deviation function.

w_var for weighted variance (the square of weighted SD).

w_mean for weighted means.

describe for comprehensive descriptive statistics including SD.

Other weighted_statistics: w_iqr(), w_kurtosis(), w_mean(), w_median(), w_modus(), w_quantile(), w_range(), w_se(), w_skew(), w_var()

Examples

# Load required packages and data
library(dplyr)
data(survey_data)

# Basic weighted standard deviation
survey_data %>% w_sd(age, weights = sampling_weight)
#> 
#> Weighted Standard Deviation Statistics
#> --------------------------------------
#> Weights: sampling_weight
#> 
#>   -------------------------------
#>   Variable      SD     N  Missing
#>   -------------------------------
#>   age       17.084  2516        0
#>   -------------------------------

# Multiple variables
survey_data %>% w_sd(age, income, life_satisfaction, weights = sampling_weight)
#> 
#> Weighted Standard Deviation Statistics
#> --------------------------------------
#> Weights: sampling_weight
#> 
#>   ------------------------------------------
#>   Variable                 SD     N  Missing
#>   ------------------------------------------
#>   age                  17.084  2516        0
#>   income             1423.966  2201      315
#>   life_satisfaction     1.152  2437       79
#>   ------------------------------------------

# Grouped data
survey_data %>% group_by(region) %>% w_sd(age, weights = sampling_weight)
#> 
#> Weighted Standard Deviation Statistics
#> --------------------------------------
#> Weights: sampling_weight
#> 
#> Group: region = East
#> --------------------
#> 
#>   ------------------------------
#>   Variable      SD    N  Missing
#>   ------------------------------
#>   age       17.595  509        0
#>   ------------------------------
#> 
#> Group: region = West
#> --------------------
#> 
#>   -------------------------------
#>   Variable      SD     N  Missing
#>   -------------------------------
#>   age       16.927  2007        0
#>   -------------------------------

# In summarise context
survey_data %>% summarise(sd_age = w_sd(age, weights = sampling_weight))
#> # A tibble: 1 × 1
#>   sd_age
#>    <dbl>
#> 1   17.1

# Unweighted (for comparison)
survey_data %>% w_sd(age)
#> 
#> Standard Deviation Statistics
#> -----------------------------
#> 
#>   -------------------------------
#>   Variable      SD     N  Missing
#>   -------------------------------
#>   age       16.976  2500        0
#>   -------------------------------