A descriptive measure is a numerical value which summarises a set of data. We represent the number of patients in a sample by the letter n. We often represent the individual values of a variable with a small letter x. The value for patient 1 would be x1, the value for patient 2 would be x2. In a sample of n patients, the value for patient n would be xn.
These give the location of the centre of the data. Representative measures are mean, median, and mode. They are referred to as ‘average’ values – i.e. ‘something in the middle’.
The sample mean (\(\bar{x}\)) is calculated as:
\[\text{mean} = \frac{\sum x}{n}\]
Where Σx = x1 + x2 + x3 + … + xn is the sum of all values across the subjects, and n is the total number of subjects.
The sample median is the middle value when the observations are ranked from lowest to highest. If n is even, it is the mean of the middle two values.
Its interpretation is that 50% of data values are above the median; 50% are below the median.
Quartiles (Q1, Q2, and Q3) – when the observations are ranked from lowest to highest the quartiles divide a set of data into four parts of equal frequency:
The sample mode is the most frequently occurring value. This term is seldom used.
Knowing the “average” value of data is not very informative by itself. We also need to know how “concentrated” or “spread out” the data are. That is, we need to know something about the “variability” of the data.
Measures of dispersion are ways of quantifying this numerically. They describe the degree to which the data vary about their average value, their scatter or spread. Representative measures: range, standard deviation, and interquartile range.
Range is simply the difference between the smallest (minimum) and largest (maximum) values in the sample.
Standard deviation (SD or sd) is a measure of how far away observations are from the sample mean.
It is calculated as:
\[SD = \sqrt{\frac{\sum(x - \text{mean})^2}{n-1}}\]
Which is the square root of the sum of squared differences from the sample mean divided by (n-1).
Interquartile range (IQR) is simply the lower quartile and upper quartile (Q1, Q3). Sometimes it is expressed as the value of Q3 - Q1.
10 patients with lung cancer had a pretreatment PET scan and SUVmax was measured.
# Original data
suv_data <- c(1.8, 8.9, 2.7, 9.4, 5.4, 16.0, 5.8, 17.9, 13.1, 6.6)
tibble(
Patient = 1:10,
`SUVmax` = suv_data
) |>
kable() |>
kable_styling(bootstrap_options = c("striped", "hover"))| Patient | SUVmax |
|---|---|
| 1 | 1.8 |
| 2 | 8.9 |
| 3 | 2.7 |
| 4 | 9.4 |
| 5 | 5.4 |
| 6 | 16.0 |
| 7 | 5.8 |
| 8 | 17.9 |
| 9 | 13.1 |
| 10 | 6.6 |
# Calculate mean
sum_values <- sum(suv_data)
n <- length(suv_data)
mean_suv <- sum_values / nThe mean SUVmax = (1.8 + 8.9 + 2.7 + 9.4 + 5.4 + 16.0 + 5.8 + 17.9 + 13.1 + 6.6) / 10 = 8.76
The calculation of the standard deviation involves:
Step 1: Subtracting the mean SUVmax from each observation:
deviations <- suv_data - mean_suv
tibble(
Patient = 1:10,
SUVmax = suv_data,
`Deviation (x - mean)` = round(deviations, 1)
) |>
kable() |>
kable_styling(bootstrap_options = c("striped", "hover"))| Patient | SUVmax | Deviation (x - mean) |
|---|---|---|
| 1 | 1.8 | -7.0 |
| 2 | 8.9 | 0.1 |
| 3 | 2.7 | -6.1 |
| 4 | 9.4 | 0.6 |
| 5 | 5.4 | -3.4 |
| 6 | 16.0 | 7.2 |
| 7 | 5.8 | -3.0 |
| 8 | 17.9 | 9.1 |
| 9 | 13.1 | 4.3 |
| 10 | 6.6 | -2.2 |
Step 2: Squaring these values:
squared_deviations <- deviations^2
sum_squared <- sum(squared_deviations)Squared deviations: 48.44, 0.02, 36.72, 0.41, 11.29, 52.42, 8.76, 83.54, 18.84, 4.67
Step 3: Summing these values together: 265.1
Step 4: Dividing by the number of observations minus 1:
\[\frac{265.1}{(10-1)} = 29.46\]
Step 5: Taking the square root:
\[\text{standard deviation} = \sqrt{29.46} = 5.43\]
Ordering the data from lowest to highest allows identification of the median and quartiles.
Ordered data: 1.8, 2.7, 5.4, 5.8, 6.6, 8.9, 9.4, 13.1, 16.0, 17.9
ordered_data <- sort(suv_data)
median_suv <- median(suv_data)
q1 <- quantile(suv_data, 0.25)
q3 <- quantile(suv_data, 0.75)
min_val <- min(suv_data)
max_val <- max(suv_data)
range_val <- max_val - min_val
iqr_val <- q3 - q1tibble(
Measure = c("Mean (SD)", "Median (IQR)", "Range"),
Value = c(
paste0(round(mean_suv, 2), " (", round(sd(suv_data), 2), ")"),
paste0(median_suv, " (", q1, "-", q3, ")"),
paste0(min_val, " to ", max_val)
)
) |>
kable() |>
kable_styling(bootstrap_options = c("striped", "hover"))| Measure | Value |
|---|---|
| Mean (SD) | 8.76 (5.43) |
| Median (IQR) | 7.75 (5.5-12.175) |
| Range | 1.8 to 17.9 |
Remember to report means with standard deviations (8.76 (SD 5.43)), and medians with interquartile ranges (7.75 (IQR 5.4-13.1)).
Do not mix means with the interquartile range or medians with standard deviations.
set.seed(42)
# Normal data
normal_data <- tibble(
value = rnorm(100, mean = 50, sd = 10),
type = "Symmetric distribution"
)
# Skewed data with outlier
skewed_data <- tibble(
value = c(rnorm(95, mean = 50, sd = 10), 120, 130, 140, 150, 160),
type = "With outliers"
)
combined <- bind_rows(normal_data, skewed_data)
ggplot(combined, aes(x = value)) +
geom_histogram(bins = 20, fill = "#3498db", alpha = 0.7) +
geom_vline(data = combined |> group_by(type) |> summarise(m = mean(value)),
aes(xintercept = m), colour = "#e74c3c", linewidth = 1, linetype = "dashed") +
geom_vline(data = combined |> group_by(type) |> summarise(m = median(value)),
aes(xintercept = m), colour = "#27ae60", linewidth = 1) +
facet_wrap(~type, scales = "free_x") +
labs(x = "Value", y = "Frequency",
caption = "Red dashed = mean, Green solid = median") +
theme_minimal(base_size = 12)
The sample mode, the sample median and the sample mean have corresponding population measures. That is, we assume that the variable in question has a population mode, population median, population mean, which are all unknown. The sample mode, the sample median and the sample mean are used to estimate the values of these corresponding unknown population parameters.
| Measure | Description | When to Use |
|---|---|---|
| Mean | Sum of values divided by n | Symmetric numeric data |
| Median | Middle value when ordered | Skewed numeric data |
| Mode | Most frequent value | Categorical data |
| Range | Maximum minus minimum | Quick measure of spread |
| Standard deviation | Average distance from mean | Symmetric numeric data |
| Interquartile range | Q3 minus Q1 | Skewed numeric data |