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

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

1import numpy as np
2 
3data = np.array([12.5, 18.2, 29.4, 15.1, 22.8, 31.0, 19.6])
4 
5print("Count:              ", data.size)
6print("Sum:                ", np.sum(data))
7print("Product:            ", np.prod(data))
8print("Minimum:            ", np.min(data))
9print("Maximum:            ", np.max(data))
10print("Mean (Average):     ", np.mean(data))
11print("Median (50th %ile): ", np.median(data))
12print("Variance:           ", np.var(data))
13print("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:

1samples = np.array([10.0, 12.0, 23.0, 23.0, 16.0, 23.0, 21.0, 16.0])
2 
3# Population standard deviation (divide by N):
4pop_std = np.std(samples, ddof=0)
5print(f"Population Std (ddof=0): {pop_std:.4f}")
6 
7# Unbiased sample standard deviation (divide by N - 1):
8sample_std = np.std(samples, ddof=1)
9print(f"Sample Std (ddof=1):     {sample_std:.4f}")

Output:

1Population Std (ddof=0): 4.8974
2Sample 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:

1# Daily stock price percentage returns
2daily_returns = np.array([0.02, -0.01, 0.03, 0.015, -0.005])
3 
4# Cumulative sum (running total)
5cumulative_gains = np.cumsum(daily_returns)
6print("Cumulative gains:", cumulative_gains)
7 
8# Compound growth: (1 + r1) * (1 + r2) * ...
9growth_factors = 1.0 + daily_returns
10wealth_index = np.cumprod(growth_factors)
11print("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:

1dirty_data = np.array([10.0, 25.0, np.nan, 40.0, 15.0])
2 
3print("Standard sum:", np.sum(dirty_data))    # Output: nan
4print("Standard mean:", np.mean(dirty_data))  # Output: nan

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

1print("NaN-safe sum:  ", np.nansum(dirty_data))   # 90.0
2print("NaN-safe mean: ", np.nanmean(dirty_data))  # 90 / 4 = 22.5
3print("NaN-safe std:  ", np.nanstd(dirty_data))   # calculates over 4 valid items
4print("NaN-safe min:  ", np.nanmin(dirty_data))   # 10.0

Interactive Quiz• Knowledge Check

Multiple Choice Questions

Question 1

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

Question 2

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

Question 3

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

Question 4

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

Question 5

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

Next Lesson

Directional Aggregations with the axis Parameter

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Universal Functions (ufuncs), Trigonometry & Logarithms	Directional Aggregations with the axis Parameter

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Directional Aggregations with the axis Parameter

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

Beginner12 min read•Updated: Jul 11, 2026

Study Materials Cheatsheet

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

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

1. Core Summary Functions

1import numpy as np
2 
3data = np.array([12.5, 18.2, 29.4, 15.1, 22.8, 31.0, 19.6])
4 
5print("Count:              ", data.size)
6print("Sum:                ", np.sum(data))
7print("Product:            ", np.prod(data))
8print("Minimum:            ", np.min(data))
9print("Maximum:            ", np.max(data))
10print("Mean (Average):     ", np.mean(data))
11print("Median (50th %ile): ", np.median(data))
12print("Variance:           ", np.var(data))
13print("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:

1samples = np.array([10.0, 12.0, 23.0, 23.0, 16.0, 23.0, 21.0, 16.0])
2 
3# Population standard deviation (divide by N):
4pop_std = np.std(samples, ddof=0)
5print(f"Population Std (ddof=0): {pop_std:.4f}")
6 
7# Unbiased sample standard deviation (divide by N - 1):
8sample_std = np.std(samples, ddof=1)
9print(f"Sample Std (ddof=1):     {sample_std:.4f}")

Output:

1Population Std (ddof=0): 4.8974
2Sample 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:

1# Daily stock price percentage returns
2daily_returns = np.array([0.02, -0.01, 0.03, 0.015, -0.005])
3 
4# Cumulative sum (running total)
5cumulative_gains = np.cumsum(daily_returns)
6print("Cumulative gains:", cumulative_gains)
7 
8# Compound growth: (1 + r1) * (1 + r2) * ...
9growth_factors = 1.0 + daily_returns
10wealth_index = np.cumprod(growth_factors)
11print("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:

1dirty_data = np.array([10.0, 25.0, np.nan, 40.0, 15.0])
2 
3print("Standard sum:", np.sum(dirty_data))    # Output: nan
4print("Standard mean:", np.mean(dirty_data))  # Output: nan

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

1print("NaN-safe sum:  ", np.nansum(dirty_data))   # 90.0
2print("NaN-safe mean: ", np.nanmean(dirty_data))  # 90 / 4 = 22.5
3print("NaN-safe std:  ", np.nanstd(dirty_data))   # calculates over 4 valid items
4print("NaN-safe min:  ", np.nanmin(dirty_data))   # 10.0

Interactive Quiz• Knowledge Check

Multiple Choice Questions

Question 1

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

Question 2

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

Question 3

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

Question 4

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

Question 5

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

Next Lesson

Directional Aggregations with the axis Parameter

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

Related Lessons

Previous Lesson	Next Lesson
Universal Functions (ufuncs), Trigonometry & Logarithms	Directional Aggregations with the axis Parameter

Practice Quiz

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

Previous Next

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

1. Core Summary Functions

2. Population vs Sample Variance: The ddof Parameter

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

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

Multiple Choice Questions

Directional Aggregations with the axis Parameter

Related Lessons

Practice Quiz

Loading Page Content...

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

1. Core Summary Functions

2. Population vs Sample Variance: The ddof Parameter

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

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

Multiple Choice Questions

Directional Aggregations with the axis Parameter

Related Lessons

Practice Quiz

2. Population vs Sample Variance: The `ddof` Parameter

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

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

2. Population vs Sample Variance: The `ddof` Parameter

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

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