Aggregate Statistics (sum, mean, median, std, var)0%
Directional Aggregations with the axis Parameter

Aggregate Statistics (sum, mean, median, std, var)

Beginner12 min read•Updated: Jul 11, 2026
Study Materials

Summary Statistics: sum(), prod(), mean(), std(), var()

Descriptive statistics summarize, describe, and reveal properties of numerical datasets. In data science, feature engineering, and statistical modeling, calculating aggregates—such as totals, averages, standard deviations, and variances—is a fundamental first step.

NumPy provides highly optimized aggregation functions implemented in C, equipped with options for numerical stability and degree-of-freedom adjustments.

1. Core Summary Functions

Python
import numpy as np
 
data = np.array([12.5, 18.2, 29.4, 15.1, 22.8, 31.0, 19.6])
 
print("Count: ", data.size)
print("Sum: ", np.sum(data))
print("Product: ", np.prod(data))
print("Minimum: ", np.min(data))
print("Maximum: ", np.max(data))
print("Mean (Average): ", np.mean(data))
print("Median (50th %ile): ", np.median(data))
print("Variance: ", np.var(data))
print("Standard Deviation: ", np.std(data))

2. Population vs Sample Variance: The ddof Parameter

A critical statistical distinction often missed by beginners is Degrees of Freedom:

$$\sigma^2_{\text{pop}} = \frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2 \qquad s^2_{\text{sample}} = \frac{1}{N - 1} \sum_{i=1}^N (x_i - \bar{x})^2$$

  • In NumPy, by default, np.var() and np.std() set ddof=0 (Population variance, dividing by $N$).
  • In Pandas, by default, .var() and .std() set ddof=1 (Sample variance, dividing by $N - 1$, Bessel's correction).

To match unbiased sample statistics in academic research or Pandas:

Python
samples = np.array([10.0, 12.0, 23.0, 23.0, 16.0, 23.0, 21.0, 16.0])
 
# Population standard deviation (divide by N):
pop_std = np.std(samples, ddof=0)
print(f"Population Std (ddof=0): {pop_std:.4f}")
 
# Unbiased sample standard deviation (divide by N - 1):
sample_std = np.std(samples, ddof=1)
print(f"Sample Std (ddof=1): {sample_std:.4f}")

Output:

Output
Population Std (ddof=0): 4.8974
Sample Std (ddof=1): 5.2372

3. Cumulative Aggregations: cumsum() and cumprod()

In time-series analysis and financial quantitative modeling, running totals and compound returns are computed via cumulative functions:

Python
# Daily stock price percentage returns
daily_returns = np.array([0.02, -0.01, 0.03, 0.015, -0.005])
 
# Cumulative sum (running total)
cumulative_gains = np.cumsum(daily_returns)
print("Cumulative gains:", cumulative_gains)
 
# Compound growth: (1 + r1) * (1 + r2) * ...
growth_factors = 1.0 + daily_returns
wealth_index = np.cumprod(growth_factors)
print("Wealth growth index:", wealth_index)

4. Handling Missing Data with nanmean, nansum, etc.

If an array contains even a single np.nan (Not a Number), standard aggregation functions return nan by default:

Python
dirty_data = np.array([10.0, 25.0, np.nan, 40.0, 15.0])
 
print("Standard sum:", np.sum(dirty_data)) # Output: nan
print("Standard mean:", np.mean(dirty_data)) # Output: nan

NumPy includes a dedicated family of NaN-safe functions that automatically ignore missing values during computation:

Python
print("NaN-safe sum: ", np.nansum(dirty_data)) # 90.0
print("NaN-safe mean: ", np.nanmean(dirty_data)) # 90 / 4 = 22.5
print("NaN-safe std: ", np.nanstd(dirty_data)) # calculates over 4 valid items
print("NaN-safe min: ", np.nanmin(dirty_data)) # 10.0

Multiple Choice Questions

1. What is the default value of the Delta Degrees of Freedom (ddof) parameter in np.std() and np.var()?

A. 1 B. 0 C. 2 D. None Answer: B Explanation: NumPy defaults to ddof=0 (population variance, dividing by $N$), unlike Pandas which defaults to ddof=1 (sample variance with Bessel's correction).

2. If an array contains [5.0, 15.0, np.nan], what does np.mean(arr) evaluate to?

A. 10.0 B. 6.67 C. np.nan D. It raises a ValueError Answer: C Explanation: Standard aggregation functions propagate NaNs. Any arithmetic operation involving NaN produces NaN. To ignore NaNs, use np.nanmean().

3. Which NumPy function computes the running cumulative sum of an array?

A. np.sum_all() B. np.cumsum() C. np.running_sum() D. np.add_reduce() Answer: B Explanation: np.cumsum() computes the cumulative sum of elements along a given axis, returning an array of the same size where each position is the sum of all preceding elements.

4. Given data = np.array([10, 20, 30]), what is the result of np.cumprod(data)?

A. [10, 200, 6000] B. 6000 C. [10, 30, 60] D. [10, 100, 1000] Answer: A Explanation: np.cumprod calculates running products: position 0 is 10; position 1 is $10 imes 20 = 200$; position 2 is $200 imes 30 = 6000$.

5. How can you calculate the unbiased sample variance (dividing by $N-1$) of an array x in NumPy?

A. np.var(x) B. np.var(x, ddof=1) C. np.sample_var(x) D. np.var(x) / (len(x) - 1) Answer: B Explanation: Setting ddof=1 in np.var() applies Bessel's correction, dividing the sum of squared deviations by $N - 1$ to calculate unbiased sample variance.

Next Lesson

Directional Aggregations with the axis Parameter

Continue learning with hands-on practice, examples, and exercises in the upcoming topic.

Related Lessons

Previous LessonNext Lesson
Universal Functions (ufuncs), Trigonometry & LogarithmsDirectional Aggregations with the axis Parameter

Practice Quiz

Test your understanding of this lesson with 5 questions. Each question has one correct answer.

PreviousNext
PrevNext