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Pearson correlation coefficient
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Not to be confused with Coefficient of determination.
In statistics, the Pearson correlation coefficient (PCC, pronounced /ˈpɪərsən/), also referred to as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC) or the bivariate correlation,[1] is a measure of the linear correlation between two variables X and Y. It has a value between +1 and −1, where 1 is total positive linear correlation, 0 is no linear correlation, and −1 is total negative linear correlation. It is widely used in the sciences. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s.[2][3][4]
Examples of scatter diagrams with different values of correlation coefficient (ρ)
Several sets of (x, y) points, with the correlation coefficient of x and y for each set. Note that the correlation reflects the non-linearity and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero.
Contents
[hide]
1 Definition
1.1 For a population
1.2 For a sample
2 Mathematical properties
3 Interpretation
3.1 Geometric interpretation
3.2 Interpretation of the size of a correlation
4 Inference
4.1 Using a permutation test
4.2 Using a bootstrap
4.3 Testing using Student's t-distribution
4.4 Using the exact distribution
4.5 Using the Fisher transformation
5 In least squares regression analysis
6 Sensitivity to the data distribution
6.1 Existence
6.2 Sample size
6.3 Robustness
7 Variants
7.1 Adjusted correlation coefficient
7.2 Weighted correlation coefficient
7.3 Reflective correlation coefficient
7.4 Scaled correlation coefficient
7.5 Pearson's distance
7.6 Circular correlation coefficient
7.7 Partial correlation
8 Decorrelation of n random variables
9 See also
10 References
11 External links
Definition[edit]
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
For a population[edit]
Pearson's correlation coefficient when applied to a population is commonly represented by the Greek letter ρ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient. The formula for ρ[5] is:
ρ X , Y = cov  ( X , Y ) σ X σ Y {\displaystyle \rho _{X,Y}={\frac {\operatorname {cov} (X,Y)}{\sigma _{X}\sigma _{Y}}}}
where:
cov {\displaystyle \operatorname {cov} } is the covariance
σ X {\displaystyle \sigma _{X}} is the standard deviation of X {\displaystyle X}
σ Y {\displaystyle \sigma _{Y}} is the standard deviation of Y {\displaystyle Y}
The formula for ρ can be expressed in terms of mean and expectation. Since
cov  ( X , Y ) = E  [ ( X − μ X ) ( Y − μ Y ) ] , {\displaystyle \operatorname {cov} (X,Y)=\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})],} [5]
the formula for ρ can also be written as
ρ X , Y = E  [ ( X − μ X ) ( Y − μ Y ) ] σ X σ Y {\displaystyle \rho _{X,Y}={\frac {\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})]}{\sigma _{X}\sigma _{Y}}}}
where:
cov {\displaystyle \operatorname {cov} } and σ X {\displaystyle \sigma _{X}} are defined as above
μ X {\displaystyle \mu _{X}} is the mean of X {\displaystyle X}
E {\displaystyle \operatorname {E} } is the expectation.
The formula for ρ can be expressed in terms of uncentered moments. Since
μ X = E  [ X ] {\displaystyle \mu _{X}=\operatorname {E} [X]}
μ Y = E  [ Y ] {\displaystyle \mu _{Y}=\operatorname {E} [Y]}
σ X 2 = E  [ ( X − E  [ X ] ) 2 ] = E  [ X 2 ] − [ E  [ X ] ] 2 {\displaystyle \sigma _{X}^{2}=\operatorname {E} [(X-\operatorname {E} [X])^{2}]=\operatorname {E} [X^{2}]-[\operatorname {E} [X]]^{2}}
σ Y 2 = E  [ ( Y − E  [ Y ] ) 2 ] = E  [ Y 2 ] − [ E  [ Y ] ] 2 {\displaystyle \sigma _{Y}^{2}=\operatorname {E} [(Y-\operatorname {E} [Y])^{2}]=\operatorname {E} [Y^{2}]-[\operatorname {E} [Y]]^{2}}
E  [ ( X − μ X ) ( Y − μ Y ) ] = E  [ ( X − E  [ X ] ) ( Y − E  [ Y ] ) ] = E  [ X Y ] − E  [ X ] E  [ Y ] , {\displaystyle \operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})]=\operatorname {E} [(X-\operatorname {E} [X])(Y-\operatorname {E} [Y])]=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y],\,}
the formula for ρ can also be written as
ρ X , Y = E  [ X Y ] − E  [ X ] E  [ Y ] E  [ X 2 ] − [ E  [ X ] ] 2 E  [ Y 2 ] − [ E  [ Y ] ] 2 . {\displaystyle \rho _{X,Y}={\frac {\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y]}{{\sqrt {\operatorname {E} [X^{2}]-[\operatorname {E} [X]]^{2}}}~{\sqrt {\operatorname {E} [Y^{2}]-[\operatorname {E} [Y]]^{2}}}}}.}
For a sample[edit]
Pearson's correlation coefficient when applied to a sample is commonly represented by the letter r and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient. We can obtain a formula for r by substituting estimates of the covariances and variances based on a sample into the formula above. So if we have one dataset {x1,...,xn} containing n values and another dataset {y1,...,yn} containing n values then that formula for r is:
r = ∑ i = 1 n ( x i − x ¯ ) ( y i − y ¯ ) ∑ i = 1 n ( x i − x ¯ ) 2 ∑ i = 1 n ( y i − y ¯ ) 2 {\displaystyle r={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}}}}}
where:
n {\displaystyle n} is the sample size
x i , y i {\displaystyle x_{i},y_{i}} are the single samples indexed with i
x ¯ = 1 n ∑ i = 1 n x i {\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}} (the sample mean); and analogously for y ¯ {\displaystyle {\bar {y}}}
Rearranging gives us this formula for r:
r = r x y = n ∑ x i y i − ∑ x i ∑ y i n ∑ x i 2 − ( ∑ x i ) 2 n ∑ y i 2 − ( ∑ y i ) 2 . {\displaystyle r=r_{xy}={\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{{\sqrt {n\sum x_{i}^{2}-(\sum x_{i})^{2}}}~{\sqrt {n\sum y_{i}^{2}-(\sum y_{i})^{2}}}}}.}
where:
n , x i , y i {\displaystyle n,x_{i},y_{i}} are defined as above
This formula suggests a convenient single-pass algorithm for calculating sample correlations, but, depending on the numbers involved, it can sometimes be numerically unstable.
Rearranging again gives us this[5] formula for r:
r = r x y = ∑ x i y i − n x ¯ y ¯ ( ∑ x i 2 − n x ¯ 2 ) ( ∑ y i 2 − n y ¯ 2 ) . {\displaystyle r=r_{xy}={\frac {\sum x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{{\sqrt {(\sum x_{i}^{2}-n{\bar {x}}^{2})}}~{\sqrt {(\sum y_{i}^{2}-n{\bar {y}}^{2})}}}}.}
where:
n , x i , y i , x ¯ , y ¯ {\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}} are defined as above
An equivalent expression gives the formula for r as the mean of the products of the standard scores as follows:
r = r x y = 1 n − 1 ∑ i = 1 n ( x i − x ¯ s x ) ( y i − y ¯ s y ) {\displaystyle r=r_{xy}={\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)\left({\frac {y_{i}-{\bar {y}}}{s_{y}}}\right)}
where
n , x i , y i , x ¯ , y ¯ {\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}} are defined as above, and s x , s y {\displaystyle s_{x},s_{y}} are defined below
( x i − x ¯ s x ) {\displaystyle \left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)} is the standard score (and analogously for the standard score of y)
Alternative formulae for r are also available. One can use the following formula for r:
r = r x y = ∑ x i y i − n x ¯ y ¯ ( n − 1 ) s x s y {\displaystyle r=r_{xy}={\frac {\sum x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{(n-1)s_{x}s_{y}}}}
where:
n , x i , y i , x ¯ , y ¯ {\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}} are defined as above and:
s x = 1 n − 1 ∑ i = 1 n ( x i − x ¯ ) 2 {\displaystyle s_{x}={\sqrt {{\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}} (the sample standard deviation); and analogously for sy
Practical issues
Under heavy noise conditions, extracting the correlation coefficient between two sets of stochastic variables is nontrivial, in particular where Canonical Correlation Analysis reports degraded correlation values due to the heavy noise contributions. A generalization of the approach is given elsewhere.[6]
In case of missing data, Garren derived the maximum likelihood estimator.[7]
Mathematical properties[edit]
The absolute values of both the sample and population Pearson correlation coefficients are less than or equal to 1. Correlations equal to 1 or −1 correspond to data points lying exactly on a line (in the case of the sample correlation), or to a bivariate distribution entirely supported on a line (in the case of the population correlation). The Pearson correlation coefficient is symmetric: corr(X,Y) = corr(Y,X).
A key mathematical property of the Pearson correlation coefficient is that it is invariant under separate changes in location and scale in the two variables. That is, we may transform X to a + bX and transform Y to c + dY, where a, b, c, and d are constants with b, d > 0, without changing the correlation coefficient. (This holds for both the population and sample Pearson correlation coefficients.) Note that more general linear transformations do change the correlation: see § Decorrelation_of_n_random_variables for an application of this.
Interpretation[edit]
The correlation coefficient ranges from −1 to 1. A value of 1 implies that a linear equation describes the relationship between X and Y perfectly, with all data points lying on a line for which Y increases as X increases. A value of −1 implies that all data points lie on a line for which Y decreases as X increases. A value of 0 implies that there is no linear correlation between the variables.
More generally, note that (Xi − X)(Yi − Y) is positive if and only if Xi and Yi lie on the same side of their respective means. Thus the correlation coefficient is positive if Xi and Yi tend to be simultaneously greater than, or simultaneously less than, their respective means. The correlation coefficient is negative (anti-correlation) if Xi and Yi tend to lie on opposite sides of their respective means. Moreover, the stronger is either tendency, the larger is the absolute value of the correlation coefficient.
Rogers and Nicewander [8] cataloged thirteen ways of interpreting correlation:
Function of raw scores and means
Standardized covariance
Standardized slope of the regression line
Geometric mean of the two regression slopes
Square root of the ratio of two variances
Mean cross-product of standardized variables
Function of the angle between two standardized regression lines
Function of the angle between two variable vectors
Rescaled variance of the difference between standardized scores
Estimated from the balloon rule
Related to the bivariate ellipses of isoconcentration
Function of test statistics from designed experiments
Ratio of two means
Geometric interpretation[edit]
Regression lines for y = gx(x) [red] and x = gy(y) [blue]
For uncentered data, there is a relation between the correlation coefficient and the angle φ between the two regression lines, y = gx(x) and x = gy(y), obtained by regressing y on x and x on y respectively. (Here φ is measured counterclockwise within the first quadrant formed around the lines' intersection point if r > 0, or counterclockwise from the fourth to the second quadrant if r < 0.) One can show[9] that if the standard deviations are equal, then r = sec φ − tan φ, where sec and tan are trigonometric functions.
For centered data (i.e., data which have been shifted by the sample means of their respective variables so as to have an average of zero for each variable), the correlation coefficient can also be viewed as the cosine of the angle θ between the two observed vectors in N-dimensional space (for N observations of each variable)[10]:ch. 5 (as illustrated for a special case in the next paragraph).
Both the uncentered (non-Pearson-compliant) and centered correlation coefficients can be determined for a dataset. As an example, suppose five countries are found to have gross national products of 1, 2, 3, 5, and 8 billion dollars, respectively. Suppose these same five countries (in the same order) are found to have 11%, 12%, 13%, 15%, and 18% poverty. Then let x and y be ordered 5-element vectors containing the above data: x = (1, 2, 3, 5, 8) and y = (0.11, 0.12, 0.13, 0.15, 0.18).
By the usual procedure for finding the angle θ between two vectors (see dot product), the uncentered correlation coefficient is:
cos  θ = x ⋅ y ‖ x ‖ ‖ y ‖ = 2.93 103 0.0983 = 0.920814711. {\displaystyle \cos \theta ={\frac {\mathbf {x} \cdot \mathbf {y} }{\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|}}={\frac {2.93}{{\sqrt {103}}{\sqrt {0.0983}}}}=0.920814711.}
This uncentred correlation coefficient is identical with the cosine similarity. Note that the above data were deliberately chosen to be perfectly correlated: y = 0.10 + 0.01 x. The Pearson correlation coefficient must therefore be exactly one. Centering the data (shifting x by E(x) = 3.8 and y by E(y) = 0.138) yields x = (−2.8, −1.8, −0.8, 1.2, 4.2) and y = (−0.028, −0.018, −0.008, 0.012, 0.042), from which
cos  θ = x ⋅ y ‖ x ‖ ‖ y ‖ = 0.308 30.8 0.00308 = 1 = ρ x y , {\displaystyle \cos \theta ={\frac {\mathbf {x} \cdot \mathbf {y} }{\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|}}={\frac {0.308}{{\sqrt {30.8}}{\sqrt {0.00308}}}}=1=\rho _{xy},}
as expected.
Interpretation of the size of a correlation[edit]
This figure gives a sense of how the usefulness of a Pearson correlation for predicting values varies with its magnitude. Given jointly normal X, Y with correlation ρ, 1 − 1 − ρ 2 {\displaystyle 1-{\sqrt {1-\rho ^{2}}}} (plotted here as a function of ρ) is the factor by which a given prediction interval for Y may be reduced given the corresponding value of X. For example, if ρ = .5, then the 95% prediction interval of Y|X will be about 13% smaller than the 95% prediction interval of Y.
Several authors have offered guidelines for the interpretation of a correlation coefficient.[11][12] However, all such criteria are in some ways arbitrary.[12] The interpretation of a correlation coefficient depends on the context and purposes. A correlation of 0.8 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences where there may be a greater contribution from complicating factors.
Inference[edit]
A graph showing the minimum absolute value of Pearson's correlation coefficient that is significantly different from zero at the 0.05 level, for a given sample size.
Statistical inference based on Pearson's correlation coefficient often focuses on one of the following two aims:
One aim is to test the null hypothesis that the true correlation coefficient ρ is equal to 0, based on the value of the sample correlation coefficient r.
The other aim is to derive a confidence interval that, on repeated sampling, has a given probability of containing ρ.
We discuss methods of achieving one or both of these aims below.
Using a permutation test[edit]
Permutation tests provide a direct approach to performing hypothesis tests and constructing confidence intervals. A permutation test for Pearson's correlation coefficient involves the following two steps:
Using the original paired data (xi, yi), randomly redefine the pairs to create a new data set (xi, yi′), where the i′ are a permutation of the set {1,...,n}. The permutation i′ is selected randomly, with equal probabilities placed on all n! possible permutations. This is equivalent to drawing the i′ randomly "without replacement" from the set {1, ..., n}. A closely related and equally justified (bootstrapping) approach is to separately draw the i and the i′ "with replacement" from {1, ..., n};
Construct a correlation coefficient r from the randomized data.
To perform the permutation test, repeat steps (1) and (2) a large number of times. The p-value for the permutation test is the proportion of the r values generated in step (2) that are larger than the Pearson correlation coefficient that was calculated from the original data. Here "larger" can mean either that the value is larger in magnitude, or larger in signed value, depending on whether a two-sided or one-sided test is desired.
Using a bootstrap[edit]
The bootstrap can be used to construct confidence intervals for Pearson's correlation coefficient. In the "non-parametric" bootstrap, n pairs (xi, yi) are resampled "with replacement" from the observed set of n pairs, and the correlation coefficient r is calculated based on the resampled data. This process is repeated a large number of times, and the empirical distribution of the resampled r values are used to approximate the sampling distribution of the statistic. A 95% confidence interval for ρ can be defined as the interval spanning from the 2.5th to the 97.5th percentile of the resampled r values.
Testing using Student's t-distribution[edit]
For pairs from an uncorrelated bivariate normal distribution, the sampling distribution of a certain function of Pearson's correlation coefficient follows Student's t-distribution with degrees of freedom n − 2. Specifically, if the underlying variables have a bivariate normal distribution, the variable
t = r n − 2 1 − r 2 {\displaystyle t=r{\sqrt {\frac {n-2}{1-r^{2}}}}}
has a Student's t-distribution in the null case (zero correlation).[13] This also holds approximately even if the observed values are non-normal, provided sample sizes are not very small.[14] For determining the critical values for r the inverse of this transformation is also needed:
r = t n − 2 + t 2 . {\displaystyle r={\frac {t}{\sqrt {n-2+t^{2}}}}.}
Alternatively, large sample approaches can be used.
Another early paper[15] provides graphs and tables for general values of ρ, for small sample sizes, and discusses computational approaches.
Using the exact distribution[edit]
For data that follows a bivariate normal distribution, the exact density function f(r) for the sample correlation coefficient r of a normal bivariate is[16][17]
f ( r ) = ( n − 2 ) Γ ( n − 1 ) ( 1 − ρ 2 ) n − 1 2 ( 1 − r 2 ) n − 4 2 2 π Γ ( n − 1 2 ) ( 1 − ρ r ) n − 3 2 2 F 1 ( 1 2 , 1 2 ; 2 n − 1 2 ; ρ r + 1 2 ) {\displaystyle f(r)={\frac {(n-2)\,\mathbf {\Gamma } (n-1)(1-\rho ^{2})^{\frac {n-1}{2}}(1-r^{2})^{\frac {n-4}{2}}}{{\sqrt {2\pi }}\,\mathbf {\Gamma } \left(n-{\frac {1}{2}}\right)(1-\rho r)^{n-{\frac {3}{2}}}}}\,\mathbf {_{2}F_{1}} \left({\frac {1}{2}},{\frac {1}{2}};{\frac {2n-1}{2}};{\frac {\rho r+1}{2}}\right)}
where:
Γ {\displaystyle \mathbf {\Gamma } } is the gamma function,
2 F 1 ( a , b ; c ; z ) {\displaystyle \,\mathbf {_{2}F_{1}} (a,b;c;z)} is the Gaussian hypergeometric function.
In the special case when ρ = 0 {\displaystyle \,\rho =0} , the exact density function f(r) can be written as:
f ( r ) = ( 1 − r 2 ) n − 4 2 B ( 1 2 , n − 2 2 ) , {\displaystyle f(r)={\frac {(1-r^{2})^{\frac {n-4}{2}}}{\mathbf {B} \left({\frac {1}{2}},{\frac {n-2}{2}}\right)}},}
where:
B {\displaystyle \mathbf {B} } is the beta function, which is one way of writing the density of a Student's t-distribution, as above.
Using the Fisher transformation[edit]
In practice, confidence intervals and hypothesis tests relating to ρ are usually carried out using the Fisher transformation:
F ( r ) = 1 2 ln  1 + r 1 − r = artanh  ( r ) . {\displaystyle F(r)={1 \over 2}\ln {1+r \over 1-r}=\operatorname {artanh} (r).}
If F(r) is the Fisher transformation of r, and n is the sample size, then F(r) approximately follows a normal distribution with
mean = F ( ρ ) = artanh  ( ρ ) {\displaystyle {\text{mean}}=F(\rho )=\operatorname {artanh} (\rho )} and standard error SE = 1 n − 3 . {\displaystyle {\text{SE}}={\frac {1}{\sqrt {n-3}}}.}
Thus, a z-score is
z = x − mean SE = [ F ( r ) − F ( ρ 0 ) ] n − 3 {\displaystyle z={\frac {x-{\text{mean}}}{\text{SE}}}=[F(r)-F(\rho _{0})]{\sqrt {n-3}}}
under the null hypothesis of that ρ = ρ 0 {\displaystyle \rho =\rho _{0}} , given the assumption that the sample pairs are independent and identically distributed and follow a bivariate normal distribution. Thus an approximate p-value can be obtained from a normal probability table. For example, if z = 2.2 is observed and a two-sided p-value is desired to test the null hypothesis that ρ = 0 {\displaystyle \rho =0} , the p-value is 2·Φ(−2.2) = 0.028, where Φ is the standard normal cumulative distribution function.
To obtain a confidence interval for ρ, we first compute a confidence interval for F( ρ {\displaystyle \rho } ):
100 ( 1 − α ) % CI : artanh  ( ρ ) ∈ [ artanh  ( r ) ± z α / 2 S E ] {\displaystyle 100(1-\alpha )\%{\text{CI}}:\operatorname {artanh} (\rho )\in [\operatorname {artanh} (r)\pm z_{\alpha /2}SE]}

The inverse Fisher transformation bring the interval back to the correlation scale.
100 ( 1 − α ) % CI : ρ ∈ [ tanh  ( artanh  ( r ) − z α / 2 S E ) , tanh  ( artanh  ( r ) + z α / 2 S E ) ] {\displaystyle 100(1-\alpha )\%{\text{CI}}:\rho \in [\operatorname {tanh} (\operatorname {artanh} (r)-z_{\alpha /2}SE),\operatorname {tanh} (\operatorname {artanh} (r)+z_{\alpha /2}SE)]}

For example, suppose we observe r = 0.3 with a sample size of n=50, and we wish to obtain a 95% confidence interval for ρ. The transformed value is artanh(r) = 0.30952, so the confidence interval on the transformed scale is 0.30952 ± 1.96/√47, or (0.023624, 0.595415). Converting back to the correlation scale yields (0.024, 0.534).
In least squares regression analysis[edit]
For more general, non-linear dependency, see Coefficient of determination § In a non-simple linear model.
The square of the sample correlation coefficient is typically denoted r2 and is a special case of the coefficient of determination. In this case, it estimates the fraction of the variance in Y that is explained by X in a simple linear regression. So if we have the observed dataset {y1,...,yn} and the fitted dataset {f1,...,fn}, and we denote the fitted dataset {f1,...,fn} with {ŷ1,...,ŷn}, then as a starting point the total variation in the Yi around their average value can be decomposed as follows
∑ i ( Y i − Y ¯ ) 2 = ∑ i ( Y i − Y ^ i ) 2 + ∑ i ( Y ^ i − Y ¯ ) 2 , {\displaystyle \sum _{i}(Y_{i}-{\bar {Y}})^{2}=\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}+\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2},}
where the Y ^ i {\displaystyle {\hat {Y}}_{i}} are the fitted values from the regression analysis. This can be rearranged to give
1 = ∑ i ( Y i − Y ^ i ) 2 ∑ i ( Y i − Y ¯ ) 2 + ∑ i ( Y ^ i − Y ¯ ) 2 ∑ i ( Y i − Y ¯ ) 2 . {\displaystyle 1={\frac {\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}+{\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}.}
The two summands above are the fraction of variance in Y that is explained by X (right) and that is unexplained by X (left).
Next, we apply a property of least square regression models, that the sample covariance between Y ^ i {\displaystyle {\hat {Y}}_{i}} and Y i − Y ^ i {\displaystyle Y_{i}-{\hat {Y}}_{i}} is zero. Thus, the sample correlation coefficient between the observed and fitted response values in the regression can be written (calculation is under expectation, assumes Gaussian statistics)
r ( Y , Y ^ ) = ∑ i ( Y i − Y ¯ ) ( Y ^ i − Y ¯ ) ∑ i ( Y i − Y ¯ ) 2 ⋅ ∑ i ( Y ^ i − Y ¯ ) 2 = ∑ i ( Y i − Y ^ i + Y ^ i − Y ¯ ) ( Y ^ i − Y ¯ ) ∑ i ( Y i − Y ¯ ) 2 ⋅ ∑ i ( Y ^ i − Y ¯ ) 2 = ∑ i [ ( Y i − Y ^ i ) ( Y ^ i − Y ¯ ) + ( Y ^ i − Y ¯ ) 2 ] ∑ i ( Y i − Y ¯ ) 2 ⋅ ∑ i ( Y ^ i − Y ¯ ) 2 = ∑ i ( Y ^ i − Y ¯ ) 2 ∑ i ( Y i − Y ¯ ) 2 ⋅ ∑ i ( Y ^ i − Y ¯ ) 2 = ∑ i ( Y ^ i − Y ¯ ) 2 ∑ i ( Y i − Y ¯ ) 2 . {\displaystyle {\begin{aligned}r(Y,{\hat {Y}})&={\frac {\sum _{i}(Y_{i}-{\bar {Y}})({\hat {Y}}_{i}-{\bar {Y}})}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\&={\frac {\sum _{i}(Y_{i}-{\hat {Y}}_{i}+{\hat {Y}}_{i}-{\bar {Y}})({\hat {Y}}_{i}-{\bar {Y}})}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\&={\frac {\sum _{i}[(Y_{i}-{\hat {Y}}_{i})({\hat {Y}}_{i}-{\bar {Y}})+({\hat {Y}}_{i}-{\bar {Y}})^{2}]}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\&={\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sqrt {\sum _{i}(Y_{i}-{\bar {Y}})^{2}\cdot \sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}}}\\&={\sqrt {\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}}.\end{aligned}}}
Thus
r ( Y , Y ^ ) 2 = ∑ i ( Y ^ i − Y ¯ ) 2 ∑ i ( Y i − Y ¯ ) 2 {\displaystyle r(Y,{\hat {Y}})^{2}={\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}}
where
r ( Y , Y ^ ) 2 {\displaystyle r(Y,{\hat {Y}})^{2}} is the proportion of variance in Y explained by a linear function of X.
That equation can be written as:
r ( Y , Y ^ ) 2 = S S reg S S tot {\displaystyle r(Y,{\hat {Y}})^{2}={\frac {SS_{\text{reg}}}{SS_{\text{tot}}}}}
where
S S reg {\displaystyle SS_{\text{reg}}} is the regression sum of squares, also called the explained sum of squares
S S tot {\displaystyle SS_{\text{tot}}} is the total sum of squares (proportional to the variance of the data)
S S reg = ∑ i ( Y ^ i − Y ¯ ) 2 {\displaystyle SS_{\text{reg}}=\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}
S S tot = ∑ i ( Y i − Y ¯ ) 2 {\displaystyle SS_{\text{tot}}=\sum _{i}(Y_{i}-{\bar {Y}})^{2}}
Sensitivity to the data distribution[edit]
Further information: Correlation and dependence § Sensitivity to the data distribution
Existence[edit]
The population Pearson correlation coefficient is defined in terms of moments, and therefore exists for any bivariate probability distribution for which the population covariance is defined and the marginal population variances are defined and are non-zero. Some probability distributions such as the Cauchy distribution have undefined variance and hence ρ is not defined if X or Y follows such a distribution. In some practical applications, such as those involving data suspected to follow a heavy-tailed distribution, this is an important consideration. However, the existence of the correlation coefficient is usually not a concern; for instance, if the range of the distribution is bounded, ρ is always defined.
Sample size[edit]
If the sample size is moderate or large and the population is normal, then, in the case of the bivariate normal distribution, the sample correlation coefficient is the maximum likelihood estimate of the population correlation coefficient, and is asymptotically unbiased and efficient, which roughly means that it is impossible to construct a more accurate estimate than the sample correlation coefficient.
If the sample size is large and the population is not normal, then the sample correlation coefficient remains approximately unbiased, but may not be efficient.
If the sample size is large then sample correlation coefficient is a consistent estimator of the population correlation coefficient as long as the sample means, variances, and covariance are consistent (which is guaranteed when the law of large numbers can be applied).
If the sample size is small then the sample correlation coefficient r is not an unbiased estimate of ρ.[5] The adjusted correlation coefficient must be used instead: see elsewhere in this article for the definition.
Robustness[edit]
Like many commonly used statistics, the sample statistic r is not robust,[18] so its value can be misleading if outliers are present.[19][20] Specifically, the PMCC is neither distributionally robust,[citation needed] nor outlier resistant[18] (see Robust statistics#Definition). Inspection of the scatterplot between X and Y will typically reveal a situation where lack of robustness might be an issue, and in such cases it may be advisable to use a robust measure of association. Note however that while most robust estimators of association measure statistical dependence in some way, they are generally not interpretable on the same scale as the Pearson correlation coefficient.
Statistical inference for Pearson's correlation coefficient is sensitive to the data distribution. Exact tests, and asymptotic tests based on the Fisher transformation can be applied if the data are approximately normally distributed, but may be misleading otherwise. In some situations, the bootstrap can be applied to construct confidence intervals, and permutation tests can be applied to carry out hypothesis tests. These non-parametric approaches may give more meaningful results in some situations where bivariate normality does not hold. However the standard versions of these approaches rely on exchangeability of the data, meaning that there is no ordering or grouping of the data pairs being analyzed that might affect the behavior of the correlation estimate.
A stratified analysis is one way to either accommodate a lack of bivariate normality, or to isolate the correlation resulting from one factor while controlling for another. If W represents cluster membership or another factor that it is desirable to control, we can stratify the data based on the value of W, then calculate a correlation coefficient within each stratum. The stratum-level estimates can then be combined to estimate the overall correlation while controlling for W.[21]
Variants[edit]
See also: Correlation and dependence § Other measures of dependence among random variables
Variations of the correlation coefficient can be calculated for different purposes. Here are some examples.
Adjusted correlation coefficient[edit]
The sample correlation coefficient r is not an unbiased estimate of ρ. For data that follows a bivariate normal distribution, the expectation E(r) for the sample correlation coefficient r of a normal bivariate is[22]
E  [ r ] = ρ − ρ ( 1 − ρ 2 ) 2 n + ⋯ , {\displaystyle \operatorname {E} \left[r\right]=\rho -{\frac {\rho \left(1-\rho ^{2}\right)}{2n}}+\cdots ,\quad } therefore r is a biased estimator of ρ . {\displaystyle \,\rho .}
The unique minimum variance unbiased estimator radj is given by[23]
( 1 ) r adj = r 2 F 1 ( 1 2 , 1 2 ; n − 1 2 ; 1 − r 2 ) , {\displaystyle (1)\qquad r_{\text{adj}}=r\mathbf {_{2}F_{1}} \left({\frac {1}{2}},{\frac {1}{2}};{\frac {n-1}{2}};1-r^{2}\right),}
where:
r , n {\displaystyle r,n} are defined as above,
2 F 1 ( a , b ; c ; z ) {\displaystyle \,\mathbf {_{2}F_{1}} (a,b;c;z)} is the Gaussian hypergeometric function.
An approximately unbiased estimator radj can be obtained[citation needed] by truncating E[r] and solving this truncated equation:
( 2 ) r = E  [ r ] = r adj − r adj ( 1 − r adj 2 ) 2 n . {\displaystyle (2)\qquad r=\operatorname {E} [r]=r_{\text{adj}}-{\frac {r_{\text{adj}}(1-r_{\text{adj}}^{2})}{2n}}.}
The solution[citation needed] to equation (2) is:
( 3 ) r a d j = r [ 1 + 1 − r 2 2 n ] , {\displaystyle (3)\qquad r_{adj}=r\left[1+{\frac {1-r^{2}}{2n}}\right],}
where in (3):
r , n {\displaystyle r,n} are defined as above,
radj is a suboptimal estimator,[citation needed][clarification needed]
radj can also be obtained by maximizing log(f(r)),
radj has minimum variance for large values of n,
radj has a bias of order 1/(n − 1).
Another proposed[5] adjusted correlation coefficient is:[citation needed]
r adj = 1 − ( 1 − r 2 ) ( n − 1 ) ( n − 2 ) . {\displaystyle r_{\text{adj}}={\sqrt {1-{\frac {(1-r^{2})(n-1)}{(n-2)}}}}.}
Note that radj ≈ r for large values of n.
Weighted correlation coefficient[edit]
Suppose observations to be correlated have differing degrees of importance that can be expressed with a weight vector w. To calculate the correlation between vectors x and y with the weight vector w (all of length n),[24][25]
Weighted mean:
m  ( x ; w ) = ∑ i w i x i ∑ i w i . {\displaystyle \operatorname {m} (x;w)={\sum _{i}w_{i}x_{i} \over \sum _{i}w_{i}}.}
Weighted covariance
cov  ( x , y ; w ) = ∑ i w i ( x i − m  ( x ; w ) ) ( y i − m  ( y ; w ) ) ∑ i w i . {\displaystyle \operatorname {cov} (x,y;w)={\sum _{i}w_{i}(x_{i}-\operatorname {m} (x;w))(y_{i}-\operatorname {m} (y;w)) \over \sum _{i}w_{i}}.}
Weighted correlation
corr  ( x , y ; w ) = cov  ( x , y ; w ) cov  ( x , x ; w ) cov  ( y , y ; w ) . {\displaystyle \operatorname {corr} (x,y;w)={\operatorname {cov} (x,y;w) \over {\sqrt {\operatorname {cov} (x,x;w)\operatorname {cov} (y,y;w)}}}.}
Reflective correlation coefficient[edit]
The reflective correlation is a variant of Pearson's correlation in which the data are not centered around their mean values.[citation needed] The population reflective correlation is
Corr r ( X , Y ) = E [ X Y ] E X 2 ⋅ E Y 2 . {\displaystyle {\text{Corr}}_{r}(X,Y)={\frac {E[XY]}{\sqrt {EX^{2}\cdot EY^{2}}}}.}
The reflective correlation is symmetric, but it is not invariant under translation:
Corr r ( X , Y ) = Corr r ( Y , X ) = Corr r ( X , b Y ) ≠ Corr r ( X , a + b Y ) , a ≠ 0 , b > 0. {\displaystyle {\text{Corr}}_{r}(X,Y)={\text{Corr}}_{r}(Y,X)={\text{Corr}}_{r}(X,bY)\neq {\text{Corr}}_{r}(X,a+bY),\quad a\neq 0,b>0.}
The sample reflective correlation is
r r x y = ∑ x i y i ( ∑ x i 2 ) ( ∑ y i 2 ) . {\displaystyle rr_{xy}={\frac {\sum x_{i}y_{i}}{\sqrt {(\sum x_{i}^{2})(\sum y_{i}^{2})}}}.}
The weighted version of the sample reflective correlation is
r r x y , w = ∑ w i x i y i ( ∑ w i x i 2 ) ( ∑ w i y i 2 ) . {\displaystyle rr_{xy,w}={\frac {\sum w_{i}x_{i}y_{i}}{\sqrt {(\sum w_{i}x_{i}^{2})(\sum w_{i}y_{i}^{2})}}}.}
Scaled correlation coefficient[edit]
Main article: Scaled correlation
Scaled correlation is a variant of Pearson's correlation in which the range of the data is restricted intentionally and in a controlled manner to reveal correlations between fast components in time series.[26] Scaled correlation is defined as average correlation across short segments of data.
Let K {\displaystyle K} be the number of segments that can fit into the total length of the signal T {\displaystyle T} for a given scale s {\displaystyle s} :
K = round  ( T s ) . {\displaystyle K=\operatorname {round} \left({\frac {T}{s}}\right).}
The scaled correlation across the entire signals r ¯ s {\displaystyle {\bar {r}}_{s}} is then computed as
r ¯ s = 1 K ∑ k = 1 K r k , {\displaystyle {\bar {r}}_{s}={\frac {1}{K}}\sum \limits _{k=1}^{K}r_{k},}
where r k {\displaystyle r_{k}} is Pearson's coefficient of correlation for segment k {\displaystyle k} .
By choosing the parameter s {\displaystyle s} , the range of values is reduced and the correlations on long time scale are filtered out, only the correlations on short time scales being revealed. Thus, the contributions of slow components are removed and those of fast components are retained.
Pearson's distance[edit]
A distance metric for two variables X and Y known as Pearson's distance can be defined from their correlation coefficient as[27]
d X , Y = 1 − ρ X , Y . {\displaystyle d_{X,Y}=1-\rho _{X,Y}.}
Considering that the Pearson correlation coefficient falls between [−1, 1], the Pearson distance lies in [0, 2]. The Pearson distance has been used in cluster analysis and data detection for communications and storage with unknown gain and offset[28]
Circular correlation coefficient[edit]
For variables X = {x1,...,xn} and Y = {y1,...,yn} that are defined on the unit circle [0, 2π), it is possible to define a circular analog of Pearson's coefficient.[29] This is done by transforming data points in X and Y with a sine function such that the correlation coefficient is given as:
r circular = ∑ i = 1 n sin  ( x i − x ¯ ) sin  ( y i − y ¯ ) ∑ i = 1 n sin  ( x i − x ¯ ) 2 ∑ i = 1 n sin  ( y i − y ¯ ) 2 {\displaystyle r_{\text{circular}}={\frac {\sum _{i=1}^{n}\sin(x_{i}-{\bar {x}})\sin(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}\sin(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}\sin(y_{i}-{\bar {y}})^{2}}}}}}
where x ¯ {\displaystyle {\bar {x}}} and y ¯ {\displaystyle {\bar {y}}} are the circular means of X and Y. This measure can be useful in fields like meteorology where the angular direction of data is important.
Partial correlation[edit]
Main article: Partial correlation
If a population or data-set is characterized by more than two variables, a partial correlation coefficient measures the strength of dependence between a pair of variables that is not accounted for by the way in which they both change in response to variations in a selected subset of the other variables.
Decorrelation of n random variables[edit]
Main article: Decorrelation
It is always possible to remove the correlations between all pairs of an arbitrary number of random variables by using a data transformation, even if the relationship between the variables is nonlinear. A presentation of this result for population distributions is given by Cox & Hinkley.[30]
A corresponding result exists for reducing the sample correlations to zero. Suppose a vector of n random variables is observed m times. Let X be a matrix where X i , j {\displaystyle X_{i,j}} is the jth variable of observation i. Let Z m , m {\displaystyle Z_{m,m}} be an m by m square matrix with every element 1. Then D is the data transformed so every random variable has zero mean, and T is the data transformed so all variables have zero mean and zero correlation with all other variables – the sample correlation matrix of T will be the identity matrix. This has to be further divided by the standard deviation to get unit variance. The transformed variables will be uncorrelated, even though they may not be independent.
D = X − 1 m Z m , m X {\displaystyle D=X-{\frac {1}{m}}Z_{m,m}X}

T = D ( D T D ) − 1 2 , {\displaystyle T=D(D^{\mathsf {T}}D)^{-{\frac {1}{2}}},}
where an exponent of −1/2 represents the matrix square root of the inverse of a matrix. The correlation matrix of T will be the identity matrix. If a new data observation x is a row vector of n elements, then the same transform can be applied to x to get the transformed vectors d and t:
d = x − 1 m Z 1 , m X , {\displaystyle d=x-{\frac {1}{m}}Z_{1,m}X,}

t = d ( D T D ) − 1 2 . {\displaystyle t=d(D^{\mathsf {T}}D)^{-{\frac {1}{2}}}.}
This decorrelation is related to principal components analysis for multivariate data.
See also[edit]
Statistics portal
Association (statistics)
Correlation and dependence
Disattenuation
Distance correlation
Maximal information coefficient
Multiple correlation
Normally distributed and uncorrelated does not imply independent
Partial correlation
Quadrant count ratio
RV coefficient
Spearman's rank correlation coefficient
References[edit]
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As early as 1877, Galton was using the term "reversion" and the symbol "r" for what would become "regression". F. Galton (5, 12, 19 April 1877) "Typical laws of heredity," Nature, 15 (388, 389, 390) : 492–495 ; 512–514 ; 532–533. In the "Appendix" on page 532, Galton uses the term "reversion" and the symbol r.
(F. Galton) (24 September 1885), "The British Association: Section II, Anthropology: Opening address by Francis Galton, F.R.S., etc., President of the Anthropological Institute, President of the Section," Nature, 32 (830) : 507–510.
Galton, F. (1886) "Regression towards mediocrity in hereditary stature," Journal of the Anthropological Institute of Great Britain and Ireland, 15 : 246–263.
Jump up ^ Karl Pearson (20 June 1895) "Notes on regression and inheritance in the case of two parents," Proceedings of the Royal Society of London, 58 : 240–242.
Jump up ^ Stigler, Stephen M. (1989). "Francis Galton's Account of the Invention of Correlation". Statistical Science. 4 (2): 73–79. doi:10.1214/ss/1177012580. JSTOR 2245329.
^ Jump up to: a b c d e Real Statistics Using Excel: Correlation: Basic Concepts, retrieved 2015-02-22
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Jump up ^ Garren, Steven T (15 June 1998). "Maximum likelihood estimation of the correlation coefficient in a bivariate normal model with missing data". Statistics & Probability Letters. Elsevier. 38 (3): 281–288. doi:10.1016/S0167-7152(98)00035-2. Retrieved 5 December 2015.
Jump up ^ Rogers and Nicewander (1988). "Thirteen Ways to Look at the Correlation Coefficient" (PDF). The American Statistician. 42 (1): 59–66.
Jump up ^ Schmid Jr., John (December 1947). "The Relationship between the Coefficient of Correlation and the Angle Included between Regression Lines". The Journal of Educational Research. 41 (4). JSTOR 27528906.
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Jump up ^ Rahman, N. A. (1968) A Course in Theoretical Statistics, Charles Griffin and Company, 1968
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Jump up ^ Soper, H. E.; Young, A. W.; Cave, B. M.; Lee, A.; Pearson, K. (1917). "On the distribution of the correlation coefficient in small samples. Appendix II to the papers of "Student" and R. A. Fisher. A co-operative study". Biometrika. 11: 328–413. doi:10.1093/biomet/11.4.328.
Jump up ^ Kenney, J. F. and Keeping, E. S., Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.
Jump up ^ W., Weisstein, Eric. "Correlation Coefficient--Bivariate Normal Distribution". mathworld.wolfram.com.
^ Jump up to: a b Wilcox, Rand R. (2005). Introduction to robust estimation and hypothesis testing. Academic Press.
Jump up ^ Devlin, Susan J; Gnanadesikan, R; Kettenring J.R. (1975). "Robust Estimation and Outlier Detection with Correlation Coefficients". Biometrika. 62 (3): 531–545. doi:10.1093/biomet/62.3.531. JSTOR 2335508.
Jump up ^ Huber, Peter. J. (2004). Robust Statistics. Wiley. [page needed]
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Jump up ^ Hotelling, H. (1953). "New Light on the Correlation Coefficient and its Transforms". Journal of the Royal Statistical Society. Series B (Methodological). 15 (2): 193–232. JSTOR 2983768.
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Jump up ^ "Re: Compute a weighted correlation". sci.tech-archive.net.
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Jump up ^ Nikolić, D; Muresan, RC; Feng, W; Singer, W (2012). "Scaled correlation analysis: a better way to compute a cross-correlogram" (PDF). European Journal of Neuroscience: 1–21. doi:10.1111/j.1460-9568.2011.07987.x.
Jump up ^ Fulekar (Ed.), M.H. (2009) Bioinformatics: Applications in Life and Environmental Sciences, Springer (pp. 110) ISBN 1-4020-8879-5
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Jump up ^ Cox, D.R., Hinkley, D.V. (1974) Theoretical Statistics, Chapman & Hall (Appendix 3) ISBN 0-412-12420-3
External links[edit]
Wikiversity has learning resources about Linear correlation
cocor – A free web interface and R package for the statistical comparison of two dependent or independent correlations with overlapping or non-overlapping variables
Interactive Flash simulation on the correlation of two normally distributed variables.
Correlation coefficient calculator – linear regression
"Critical values for Pearson's correlation coefficient (large table)" (PDF).
Guess the Correlation – A game where players guess how correlated two variables in a scatter plot are, in order to gain a better understanding of the concept of correlation.
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Makapansgat pebble
From Wikipedia, the free encyclopedia
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The Makapansgat pebble, or the pebble of many faces, (ca. 3,000,000 BP) is a 260-gram reddish-brown jasperite cobble with natural chipping and wear patterns that make it look like a crude rendition of a human face. The pebble is interesting in that it was found some distance from any possible natural source, associated with the bones of Australopithecus africanus in a cave in Makapansgat, South Africa.[1] Though it is definitely not a manufactured object, it has been suggested that some australopithecine might have recognized it as a symbolic face, in possibly the earliest example of symbolic thinking or aesthetic sense in the human heritage, and brought the pebble back to the cave. This would make it a candidate for the oldest known manuport.[2]
Contents
[hide]
1 Archaeological history
2 Significance
3 See also
4 References
5 External links
Archaeological history[edit]
The teacher Wilfred I. Eizman found it in the Makapansgat, a dolerite cave in the Makapan Valley north of Mokopane, Limpopo, South Africa in 1925. Almost 50 years later, Raymond Dart was the first to describe it in 1974.[3]
Significance[edit]
The Makapansgat pebble cannot be seen as art if a usual definition of the term is used, as the object was found and not made. Nevertheless that an Australopithecus may have recognized a face would reveal that the early hominid had some sort of capacity for symbolic thinking, necessary for the development of art and language.[1] If the early hominid has seen this object really as a face, or had magical speculations towards this object or just enjoyed the pebble remains unclear.
See also[edit]
Pre-historic art
Venus of Berekhat Ram
References[edit]
^ Jump up to: a b Kleiner, Fred S. (2011). Gardner's Art Through the Ages: A Global History (Enhanced Thirteenth ed.). Boston: Wadsworth. pp. 15–16. ISBN 978-0-495-79986-3.
Jump up ^ Bednarik, Robert G. "Makapansgat cobble analysed". University of Melbourne. Archived from the original on 2003-03-30. Retrieved 2010-05-14. Archived by the Internet Archive, original URI was {{cite web|url=http://sunspot.sli.unimelb.edu.au/aura/MAKAPANSGAT.htm |title=Archived copy |accessdate=2010-05-14 |deadurl=yes |archiveurl=https://web.archive.org/web/20030330065452/http://sunspot.sli.unimelb.edu.au/aura/MAKAPANSGAT.htm |archivedate=2003-03-30 |df= }}
Jump up ^ OriginsNet: Pebble of many faces
External links[edit]
Photograph of the Magapansgat pebble and article by Robert G. Bednarik
photograph of the Makapansgat pebble
Retrieved from "https://en.wikipedia.org/w/index.php?title=Makapansgat_pebble&oldid=800336112"
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