What Is a Correlation Coefficient? Positive, Negative, and No Correlation
By Markets & Investing DeskUpdated Nov 10, 2026
Definition
The correlation coefficient is a number from -1 to +1 that measures the strength and direction of the linear relationship between two variables, such as the returns of two securities.
Key takeaways
- The correlation coefficient ranges from -1 to +1: the sign gives direction, the distance from zero gives strength.
- Zero correlation means no linear relationship; two variables can still be strongly related in a curved way.
- A coefficient of +1 is perfect positive correlation, not "no correlation."
- Correlation does not imply causation, and it can change with the time period and data frequency used.
- In a portfolio, correlation affects risk (volatility), not expected return.
The correlation coefficient is a number between -1 and +1 that measures the strength and direction of the linear relationship between two variables. A value of +1 means the two move together in perfect proportion. A value of -1 means they move in exactly opposite directions. A value of 0 means no correlation: there is no straight-line relationship between them. The farther the number is from zero, in either direction, the stronger the linear relationship.
In finance, the variables are usually the returns of two securities, such as a stock and an index, two funds, or stocks and bonds. The coefficient shows whether those returns have tended to rise and fall together, and how tightly.
What the Correlation Coefficient Measures
The most widely used version is the Pearson product-moment correlation coefficient, often called Pearson’s r. It is written as r when calculated from a sample of data and as the Greek letter ρ (rho) when referring to the population value. Unless stated otherwise, “the correlation coefficient” means Pearson’s r.
Picture each pair of observations as a dot on a scatter plot, with one variable on the horizontal (x) axis and the other on the vertical (y) axis. Pearson’s r measures how closely those dots cluster around a single straight line, the line of best fit. If every dot sits exactly on an upward-sloping line, r equals +1. If every dot sits exactly on a downward-sloping line, r equals -1. If the dots form a shapeless cloud with no upward or downward tilt, r is close to 0.
The coefficient has no units. Whether you compare returns in percent, prices in dollars or temperatures in degrees, the result always lands on the same -1 to +1 scale. That is what makes it useful for comparing relationships across very different data sets.
Positive, Negative, and No Correlation
Positive Correlation
A positive correlation (r greater than 0) means that when one variable is above its average, the other tends to be above its average too, and when one is below, the other tends to be below. At r = +1, the relationship is perfectly linear: every move in one variable is matched by a proportional move in the other in the same direction.
A hypothetical illustration: if oil prices and airfares had a correlation of +0.95, the relationship would be very strong and positive. Periods of rising oil prices would line up closely with periods of rising fares, and falling oil prices with falling fares.
Negative Correlation
A negative (inverse) correlation (r less than 0) means the variables tend to move in opposite directions: when one is above its average, the other tends to be below. At r = -1, every increase in one is matched by a proportional decrease in the other.
A familiar hypothetical example is outdoor temperature and home heating bills. A coefficient of -0.96 would describe a very strong inverse relationship: as temperatures fall, heating costs rise, and as temperatures rise, heating costs fall.
No Correlation (Zero Correlation)
A correlation of 0, or very close to it, means there is no linear relationship: knowing that one variable is above its average tells you nothing, in a straight-line sense, about the other. Values roughly between -0.1 and +0.1 are often described as showing no linear relationship or only a very weak one.
The word linear matters. Zero correlation does not prove the variables are unrelated. Consider this hypothetical data set, where y is exactly x squared:
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | 4 | 1 | 0 | 1 | 4 |
Here y is completely determined by x, yet Pearson’s r is exactly 0. The products of x and y (-8, -1, 0, 1, 8) sum to zero, and so does x, so the numerator of the formula is zero. The relationship is a U-shaped curve, and a straight line cannot capture it. The upward slope on the right side cancels the downward slope on the left.
A Common Mix-Up: +1 Is Not “No Correlation”
A correlation coefficient of 1 means perfect positive correlation, the strongest possible relationship. The value that signals no linear correlation is 0. The confusion often comes from the word “independent”: statistically independent variables have zero correlation, but zero correlation alone does not guarantee independence, as the U-shaped example shows.
How Strong Is a Correlation Coefficient?
There is no universal cutoff, but a common rough scale treats the absolute value of r like this:
- 0.75 to 1.0: high degree of correlation
- 0.3 to 0.75: moderate correlation
- 0 to 0.3: weak or no correlation
The same bands apply to negative values. A coefficient of -0.8 is just as strong as +0.8; only the direction differs.
What counts as “strong” depends on the field. In physics and chemistry, researchers may require a coefficient above 0.9 or below -0.9 before treating a correlation as meaningful. In the social sciences, a threshold of 0.5 or -0.5 can be acceptable because human behavior is noisier.
Strength and reliability are separate questions. When r is calculated from a sample, whether it is statistically significant depends on a p-value, which reflects both the size of the coefficient and the number of observations. A coefficient of 0.6 from five data points can easily arise by chance; the same 0.6 from several hundred observations is far less likely to be a fluke.
Correlation Coefficient Formula
The Definition: Covariance Divided by Standard Deviations
The Pearson correlation coefficient is the covariance of two variables divided by the product of their standard deviations:
ρ(X,Y) = Cov(X,Y) / (σX × σY)
where:
- Cov(X,Y) = covariance of X and Y
- σX = standard deviation of X
- σY = standard deviation of Y
Covariance measures whether two variables move together: it is positive when they tend to be above or below their averages at the same time and negative when they tend to be on opposite sides. Its drawback is that its size is unbounded and depends on the units of the data, so a covariance of 2 or 2,000 says little on its own. Dividing by the two standard deviations, which measure how widely each variable is dispersed around its average, normalizes the result onto the -1 to +1 scale.
The Computational Formula
The same value can be calculated directly from sums of the raw data:
r = [n(Σxy) − (Σx)(Σy)] / √{[nΣx² − (Σx)²] × [nΣy² − (Σy)²]}
where n is the number of paired observations, Σxy is the sum of each x multiplied by its paired y, and Σx² and Σy² are the sums of the squared values.
Worked Example
Assume the following hypothetical monthly returns (in percent) for two funds over five months:
| Month | Fund A (x) | Fund B (y) |
|---|---|---|
| 1 | 1 | 2 |
| 2 | 2 | 1 |
| 3 | 3 | 4 |
| 4 | 4 | 3 |
| 5 | 5 | 5 |
Step 1: Compute the sums.
- n = 5
- Σx = 15, Σy = 15
- Σxy = (1×2) + (2×1) + (3×4) + (4×3) + (5×5) = 53
- Σx² = 55, Σy² = 55
Step 2: Numerator. 5 × 53 − (15 × 15) = 265 − 225 = 40
Step 3: Denominator. √[(5 × 55 − 225) × (5 × 55 − 225)] = √(50 × 50) = 50
Step 4: Divide. r = 40 / 50 = 0.8
The covariance route gives the same answer. Both funds average 3%. Multiplying each month’s deviations from the average and summing gives 8; dividing by n − 1 = 4 gives a sample covariance of 2. Each fund’s sample variance is 2.5, so each standard deviation is about 1.58. Then 2 / (1.58 × 1.58) = 2 / 2.5 = 0.8.
A coefficient of 0.8 is a strong positive correlation: in this sample, months when Fund A did better than its average were mostly months when Fund B did too. Squaring it gives r² = 0.64, meaning a straight-line fit of one fund’s returns on the other’s accounts for 64% of the variation in this small sample.
Is the Correlation Coefficient an Average?
Not of the data itself, but it can be read as one. Convert every observation into a z-score (its distance from the mean in standard deviations). Pearson’s r equals the sum of the products of each pair’s z-scores divided by n − 1, roughly the average agreement between the two standardized deviations. Pairs on the same side of their means push r up; pairs on opposite sides push it down.
How to Calculate Correlation in Practice
With real data, people use a spreadsheet, a calculator or statistics software.
In Excel or Google Sheets, place the two data series in adjacent columns and use =CORREL(array1, array2). PEARSON returns the same value.
For a correlation matrix across many series in Excel, enable the Analysis ToolPak add-in, then choose Data, Data Analysis, Correlation and enter the input range.
On a TI-84 graphing calculator, turn on DiagnosticOn from the catalog (otherwise r will not display), enter x values in L1 and y values in L2, then run STAT, CALC, 4:LinReg(ax+b) and read r along with r².
In simple two-variable analysis, correlation and simple linear regression produce the same r: the calculator gets it from a regression routine, Excel’s CORREL directly.
Use returns, not prices. When two price series both trend upward over time, their prices can show a high correlation even if their day-to-day or month-to-month movements are unrelated. Correlating periodic returns measures whether the assets actually move together from period to period.
How Investors Use the Correlation Coefficient
Diversification and Portfolio Risk
Correlation is central to diversification and to modern portfolio theory, which holds that combining assets whose returns do not move in lockstep can lower a portfolio’s volatility. The two-asset portfolio variance formula shows exactly where correlation enters:
σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂
where w is each asset’s weight, σ its standard deviation and ρ the correlation between them. Only the last term contains ρ. With ρ = +1, the portfolio’s standard deviation is simply the weighted average of the two standard deviations, and there is no diversification benefit. Any value below +1 makes portfolio volatility lower than that weighted average, and the lower the correlation, the larger the reduction. With ρ = -1, it is possible to choose weights that eliminate volatility entirely in theory.
Correlation affects risk, not expected return. A portfolio’s expected return is the weighted average of its holdings’ expected returns, and correlation does not appear in that calculation. It changes how bumpy the ride is, which in turn affects risk-adjusted measures such as the Sharpe ratio.
Investments that “offset each other” have negative correlation, not merely low correlation. A coefficient near zero means the assets move independently in a linear sense: sometimes one cushions the other, sometimes both fall together. A negative coefficient means that, over the measured period, one has tended to rise when the other fell.
A hypothetical example: a $100,000 portfolio is split 60% stocks and 40% bonds. In a strong economy with rising interest rates, stocks return 12% and bonds -2%, so the portfolio returns (0.6 × 12%) + (0.4 × -2%) = 6.4%. The next year the economy slows and rates fall; stocks return -5% and bonds 8%, so the portfolio returns 0.2%. An all-stock portfolio would swing from 12% to -5% instead. The assumed opposite moves in the two asset classes narrow the range.
Comparing Securities, Funds and Benchmarks
Correlation can be measured between any two return series: a stock and its sector, a fund and its benchmark index, two ETFs, or an asset class and the S&P 500. A fund highly correlated with its benchmark is tracking it closely; a lower correlation means other factors drove its returns. A stock’s correlation with its industry shows how much it trades with its peers versus on its own news.
Correlation is also linked to beta, the measure of a stock’s sensitivity to the market. Beta equals the correlation between the stock and the market multiplied by the ratio of the stock’s standard deviation to the market’s standard deviation. Two stocks with the same correlation to the index can have very different betas if one is much more volatile. Beta and correlation also feed into alpha, the return left after accounting for market exposure.
Factor Investing and Quantitative Strategies
Correlation calculations are used in factor investing, which builds portfolios around characteristics linked to excess returns, and by quantitative traders who use historical correlations to anticipate near-term price changes.
In pairs trading (also called divergence or convergence trading), a trader finds two securities whose returns have historically been highly correlated. When their prices drift apart more than usual, the trader buys the laggard and sells short the leader, betting the relationship will pull them back together; regression often defines the “normal” relationship. The risk is that the divergence reflects a real change, such as a business shift at one company, and the gap widens instead of closing.
Correlation vs. Related Measures
Correlation vs. covariance. Covariance is in the units of the data and has no fixed range; correlation rescales it to -1 to +1, so it can be compared across data sets.
r vs. R-squared. They are not the same. r is the correlation coefficient and shows direction and strength. R-squared, the coefficient of determination, is r squared in simple two-variable regression and shows the share of one variable’s variation explained by a linear fit on the other. Because squaring removes the sign, r = -0.8 and r = +0.8 both give R-squared of 0.64.
Correlation vs. simple linear regression. Pearson’s r measures how tightly data fit a straight line, with no assumption about which variable depends on which. Simple linear regression models a response variable (y) as a function of an explanatory variable (x) and produces a slope and intercept that can be used for prediction. Correlation does not tell you the slope: a correlation of 0.9 is consistent with a steep line or a shallow one.
Pearson vs. rank correlations. Spearman’s rank correlation and the Kendall rank correlation coefficient work on the order of the data rather than raw values. They capture monotonic relationships (consistently increasing or decreasing, even if curved) and are less sensitive to outliers. The polychoric correlation coefficient is used for ordered categorical data.
Limitations of the Correlation Coefficient
Correlation is not causation. A high coefficient shows that two variables have moved together, not that one drives the other. Both may respond to a third factor, or the pattern may be coincidence. Pearson’s r cannot distinguish dependent from independent variables.
It only captures linear relationships. As the U-shaped example showed, a strong nonlinear relationship can produce r near zero; a scatter plot reveals this quickly.
Outliers can distort it. One extreme data point far from the rest can inflate or deflate r substantially, especially in small samples.
It assumes reasonably well-behaved data. Pearson’s r is less reliable for sampled data that is far from normally distributed; rank-based methods are often used instead.
It depends on the window and frequency. A correlation describes the relationship over the specific period and interval measured. Daily, weekly and monthly returns for the same pair of assets can give different coefficients, and a relationship measured over one year may not hold the next. Negatively correlated securities, in particular, are rarely negatively correlated at all times.
FAQs
Can the correlation coefficient be greater than 1 or less than -1?
No. Because covariance can never exceed the product of the two standard deviations in absolute value, Pearson's r is mathematically bounded between -1 and +1. A result outside that range signals a calculation or data error.
The Bottom Line
Investors building diversified portfolios can use the correlation coefficient to judge whether holdings truly move independently. Lower correlation reduces portfolio volatility without changing expected return, and only negative correlation means assets tend to offset each other. Calculate it from returns rather than prices, and recheck it over different periods, since correlations shift over time and never prove that one asset drives another.
