Formulas · Published May 10, 2026 · 5-min read
The Weighted Moving Average uses linearly increasing weights — the most recent bar gets weight N, the oldest gets weight 1. Here is the formula, a worked example on NIFTY 50, and how to implement it in Excel and Python.
The Weighted Moving Average (WMA) uses a linear weight schedule — the most recent bar gets the highest weight, the oldest bar in the window gets weight 1, and every bar in between gets a weight equal to its position from oldest to newest. This guide walks through the formula, the closed-form denominator, a worked example on NIFTY 50, and implementations in Excel and Python.
For a closing-price series P and a window of N bars, the Weighted Moving Average at time t is:
N × P_t + (N−1) × P_t-1 + (N−2) × P_t-2 + ... + 1 × P_t-(N−1)
WMA_t = ───────────────────────────────────────────────────────────────────
N + (N−1) + (N−2) + ... + 1
The numerator is a weighted sum: each price multiplied by its weight, where weights run from N (newest) down to 1 (oldest). The denominator is the sum of weights.
The denominator has a clean closed form: N × (N+1) / 2, the sum of the first N positive integers.
| Window N | Denominator |
|---|---|
| 5 | 15 |
| 10 | 55 |
| 20 | 210 |
| 50 | 1,275 |
| 100 | 5,050 |
This means the most recent bar in a 10-period WMA contributes 10/55 ≈ 18.2% of the total. The oldest bar contributes 1/55 ≈ 1.8%. By contrast, in a 10-period SMA, every bar contributes exactly 10%.
Suppose the last 5 NIFTY 50 daily closes are:
| Position from oldest | Day | Close | Weight |
|---|---|---|---|
| 1 (oldest) | t−4 | 24,000 | 1 |
| 2 | t−3 | 24,100 | 2 |
| 3 | t−2 | 24,250 | 3 |
| 4 | t−1 | 24,200 | 4 |
| 5 (newest) | t | 24,400 | 5 |
Numerator: (5 × 24,400) + (4 × 24,200) + (3 × 24,250) + (2 × 24,100) + (1 × 24,000)
= 122,000 + 96,800 + 72,750 + 48,200 + 24,000
= 363,750.
Denominator: 5 + 4 + 3 + 2 + 1 = 15.
WMA = 363,750 / 15 = 24,250.
Compare to the 5-period SMA over the same five closes: (24,000 + 24,100 + 24,250 + 24,200 + 24,400) / 5 = 24,190. The WMA, at 24,250, is closer to the most recent close (24,400) because the weighting pulls it toward recent strength — exactly what WMA is designed to do.
For a 10-period WMA with closes in column A starting at A2, the formula in B11 is:
=SUMPRODUCT(A2:A11, ROW(A2:A11) − ROW(A2) + 1) / 55
How it works:
ROW(A2:A11) − ROW(A2) + 1 produces the array {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.SUMPRODUCT(A2:A11, {1, 2, ..., 10}) multiplies each close by its weight and sums.Drag the formula down. Excel automatically slides both the close range and the row-position calculation, keeping the weights 1..10 for every step.
For a 20-period WMA, change A2:A11 to A2:A21 and the divisor 55 to 210.
In pandas, rolling().apply() does the work. For a 10-period WMA:
import numpy as np
import pandas as pd
closes = pd.Series([24000, 24100, 24250, 24200, 24400, 24500, 24550, 24600, 24700, 24800, 24850, 24900])
n = 10
weights = np.arange(1, n + 1) # [1, 2, 3, ..., 10]
denominator = weights.sum() # 55
def wma_func(window):
return (window * weights).sum() / denominator
wma = closes.rolling(n).apply(wma_func, raw=True)
print(wma.tolist())
For larger datasets the raw=True flag makes pandas pass numpy arrays into the function, which is much faster than passing pandas Series.
//@version=5
indicator("Custom WMA", overlay=true)
length = input.int(20, title="Length")
wma_val = ta.wma(close, length)
plot(wma_val, color=color.purple, linewidth=2)
TradingView’s built-in ta.wma uses the linear-weight formula above. There is also ta.vwma (volume-weighted) and ta.alma (Arnaud Legoux) for alternative weighting schemes.
The conventions overlap with EMA and SMA:
A practical observation: on NIFTY 50 daily, the 9/21 WMA crossover historically flips 1–2 trading days earlier than the 9/21 EMA crossover at trend changes — but with comparable false-signal rate. WMA is essentially a slightly faster EMA with a cleaner mathematical formulation.
A useful intuition for picking the right WMA period: see how much “voice” the most recent bar has.
| N | Weight of newest bar | Weight of oldest bar |
|---|---|---|
| 5 | 33.3% | 6.7% |
| 10 | 18.2% | 1.8% |
| 20 | 9.5% | 0.5% |
| 50 | 3.9% | 0.08% |
| 200 | 1.0% | 0.005% |
For a fast WMA (5, 9, 14), the most recent bar dominates — single-bar volatility moves the WMA noticeably. For a slow WMA (50, 100, 200), recent bars are small contributions; the line is dominated by the bulk of the lookback.
This is why short WMAs are good for early signals (they “see” the latest bar clearly) and long WMAs are good for trend filters (they smooth out single-bar noise).
WMA and EMA are similar in spirit — both weight recent bars more heavily than older ones. They differ in how the weights decay:
In practice, the two indicators move very close together. WMA tends to react fractionally faster on the latest bar; EMA tends to be slightly smoother across multi-week ranges. For most traders, the choice is more about implementation preference than measurable edge.
For a side-by-side EMA vs SMA comparison on NIFTY 50 — including average lag at trend turns and 5-year forward-return profile — see our EMA vs SMA page.
The WMA formula is straightforward arithmetic — multiply each close by its position-weight, sum, divide by the closed-form denominator N(N+1)/2. The output is a moving average that responds to recent price changes faster than SMA but with a simpler, more transparent weighting scheme than EMA.
For Indian-index trend definition, WMA at 50 on daily is a sensible alternative to 50-EMA. For intraday work, 9/21 WMA crossovers fire slightly earlier than EMA equivalents at trend changes. As always, no moving average is a complete trading system — pair WMA with the regime classifier and multi-timeframe alignment tools on emaindicator.com before risking capital.