Cumulative Sum — Definition, Formula & Examples

A cumulative sum is a running total where each entry equals the sum of all previous values in the data set up to and including that point. It shows how a total grows as you move through your data from first value to last.

Given a sequence of values $x_1, x_2, \ldots, x_n$ , the cumulative sum at position $k$ is defined as $S_k = \sum_{i=1}^{k} x_i$ for $k = 1, 2, \ldots, n$ . Each term in the resulting sequence represents the partial sum of the original data through that index.

Key Formula

S_k = \sum_{i=1}^{k} x_i = x_1 + x_2 + \cdots + x_k

Where:

$S_k$ = Cumulative sum at position k
$x_i$ = The individual data value at position i
$k$ = The index up to which you are summing (1 through n)

How It Works

Start with the first value in your data set — that value is also the first cumulative sum. For each subsequent value, add it to the previous cumulative sum to get the new running total. The final cumulative sum always equals the total sum of the entire data set. Cumulative sums are especially useful when building cumulative frequency tables or cumulative relative frequency plots (ogives) in AP Statistics.

Worked Example

Problem: A teacher records quiz scores for 5 students: 8, 5, 10, 7, 10. Find the cumulative sum at each position.

Position 1: The first cumulative sum is just the first value.

S_1 = 8

Position 2: Add the second value to the previous cumulative sum.

S_2 = 8 + 5 = 13

Position 3: Continue adding each new value to the running total.

S_3 = 13 + 10 = 23

Position 4: Add the fourth value.

S_4 = 23 + 7 = 30

Position 5: Add the last value to get the overall total.

S_5 = 30 + 10 = 40

Answer: The cumulative sums are 8, 13, 23, 30, 40. The final value confirms the total of all scores is 40.

Why It Matters

Cumulative sums are the foundation of cumulative frequency distributions and ogives, both tested on the AP Statistics exam. In data analysis and quality control, cumulative sum (CUSUM) charts detect small shifts in a process mean that standard control charts might miss.

Common Mistakes

Mistake: Restarting the sum instead of carrying it forward

Correction: Each cumulative sum must include all prior values, not just the current one. Always add the new value to the previous running total, not to zero.