Moving average calculator

How do you calculate a simple moving average?

Add the last n prices and divide by n: SMA = (p1 + p2 + ... + pn) / n. Then move the window down one row and repeat. Each new price enters the average and the oldest one leaves it, which is what makes the average move.

A short example with the prices 10, 11, 12, 13 and 14 and a 3-period average:

No average exists until n prices have been entered, so a 3-period average starts on row 3. The default inputs hold 20 prices from 100.00 to 112.00 with a 5-period window: the latest SMA is (108.50 + 110.10 + 109.60 + 111.30 + 112.00) / 5 = 110.30.

Exponential moving average formula: the 2 / (n + 1) smoothing

The EMA blends each new price with the previous EMA: EMA today = a x price today + (1 - a) x EMA yesterday, with the smoothing factor a = 2 / (n + 1). For a 5-period EMA, a = 2 / 6 = 0.3333, so the newest price carries a third of the weight; for 10 periods, a = 0.1818; for 20 periods, 0.0952.

The formula needs a starting value, and the calculator offers two:

• Seed with the SMA (the default here): the EMA starts on row n, equal to the simple average of the first n prices, and the formula runs from the next row.
• Seed with the first price: the EMA starts on row 1 at that price.

In the 10 to 14 example, the SMA seed gives 11 on row 3, then 0.5 x 13 + 0.5 x 11 = 12 and 0.5 x 14 + 0.5 x 12 = 13. Seeded with the first price instead, the EMA runs 10, 10.5, 11.25, 12.125 and 13.0625. On the default 20 prices the two seeds end at 110.4768 and 110.4775, a gap that only shows at the fourth decimal. If an EMA here does not match a chart to the last decimal, the seed and the number of prices before it are the first things to check.

Weighted moving average formula

A weighted moving average gives the newest price a weight of n, the one before it n - 1, and so on down to 1 for the oldest in the window, then divides by the sum of the weights, n(n + 1) / 2. WMA = (1 x oldest + 2 x next + ... + n x newest) / (n(n + 1) / 2).

For a 5-period WMA the weights are 1, 2, 3, 4 and 5 and they sum to 15. On the default prices the latest WMA is (1 x 108.50 + 2 x 110.10 + 3 x 109.60 + 4 x 111.30 + 5 x 112.00) / 15 = 1,662.70 / 15 = 110.85. Unlike the EMA, a WMA forgets a price completely once it leaves the window.

SMA vs EMA vs WMA: how far behind each one sits

A useful way to compare them is the average age of the prices inside each one, measured in rows behind the latest price:

• SMA: the weights are equal, so the average age is (n - 1) / 2. For n = 5 that is 2 rows.
• EMA: seeded with the SMA, as the default here, the average age is exactly (n - 1) / 2, the same as the SMA, however many prices follow the seed. That holds because a = 2 / (n + 1) is the smoothing that makes (1 - a) / a equal (n - 1) / 2. Seeded with the first price, it is a little lower, (n - 1) / 2 x (1 - (1 - a)^(N - 1)) over N prices: 1.9991 rows for n = 5 on the default 20 prices.
• WMA: the linear weights pull the average age down to (n - 1) / 3, or 1.33 rows for n = 5.

On the default prices, which mostly rise, that ordering shows up in the latest values:

The WMA sits closest to the latest price in both rows. With the SMA seed the EMA and SMA share the same average age, but the EMA puts more weight on the newest prices and some on every earlier one, so the two values still differ.

Worked examples on shares in pounds and euros

• A UK share in pounds, 4-period averages. Eight closes: £4.10, £4.15, £4.08, £4.20, £4.26, £4.31, £4.28 and £4.35. The latest SMA is (4.26 + 4.31 + 4.28 + 4.35) / 4 = £4.30, and the last close is 1.16% above it. The EMA, with a = 2 / 5 = 0.4 and seeded at £4.1325, the average of the first four closes, ends at £4.2915. The WMA is £4.312.
• A share in euros, falling, 3-period averages. Six closes: €52.00, €51.40, €50.80, €51.60, €50.20 and €49.60. The latest SMA is €50.4667, the EMA €50.225 and the WMA €50.1333. The last close is 1.72% below the SMA.

In the falling series the order flips: the WMA, which leans on the newest prices, is now the lowest of the three. Prices are typed as the chart quotes them and are never converted, so a share quoted in pence works the same way and the page has no currency selector.

How the period changes the average

A longer period puts more prices into the window, so each new price moves the average less. Change the default from 5 to 10 periods and the latest SMA falls from 110.30 to 108.80, further below the latest price of 112.00, because the 10-row window reaches back to closes of 105.80 and 106.40. The EMA smoothing falls from 0.3333 to 0.1818 and the EMA from 110.48 to 108.90.

The period also sets how many prices you need before any average appears: a 20-period average needs 20 rows. Long settings such as 50 and 200 periods need that many closes, so this page, which takes up to 60 rows, covers a 50-period average but not a 200-period one; a spreadsheet handles the longer series with the formulas below.

How to calculate a moving average in Excel or Google Sheets

Put the prices in B2 to B21, oldest at the top. For a 5-period average:

• SMA in C6: =AVERAGE(B2:B6), then fill down. C21 returns 110.3 on the default prices.
• WMA in D6: =SUMPRODUCT(B2:B6,{1;2;3;4;5})/15, then fill down. D21 returns about 110.85.
• EMA with the SMA seed: in E6 enter =C6, and in E7 enter =2/(5+1)*B7+(1-2/(5+1))*E6, then fill down from E7. E21 returns about 110.48.

For the first-price seed, enter =B2 in E2 and fill the E7 style formula down from E3. Keep the period in its own cell if you want to change it in one place; the WMA weights then need rebuilding for each period.

What moving averages can and cannot tell you

Every moving average is built only from prices that already exist, so it trails the latest price by design. It can summarize where prices have been on average; it cannot say where the next price will be, and a cross of price and average says nothing certain either. This page reports the arithmetic on the rows you entered.

• Rows in the wrong order. The newest price belongs at the bottom. Reversed rows give a different answer.
• Gaps and zeros. A zero is not a price, so an empty or zero row is skipped and the note under the result says how many were skipped. The average then runs straight across the gap.
• Different seeds. An EMA seeded differently, or started from fewer prices, will not match to the last decimal.
• Mixing closes with other prices. An average of closes and an average of highs are different series.
• Comparing periods across timeframes. A 20-period average on hourly candles and on daily candles covers very different spans of time.

Glossary

Period (n): the number of prices in each average. SMA: the simple moving average, an equal-weight mean of the last n prices. EMA: the exponential moving average, which weights recent prices more and never fully drops older ones. Smoothing factor (a): the EMA's weight on the newest price, 2 / (n + 1). Seed: the value the EMA starts from. WMA: the linearly weighted moving average. Lag: how far an average trails the latest price.

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