The Pearson Product Moment Correlation Coefficient (r) measures the strength of the linear relationship between two variables. The r value is calculated using a formula involving the sum of the products of paired values and sum of squares for the two variables. r values range from +1 to -1, with values closer to these extremes indicating a stronger correlation and values closer to 0 indicating a weaker or no correlation. A positive r represents a positive correlation and a negative r represents a negative correlation. The example calculates r = 0.962 for time spent studying and test scores, indicating a strong positive correlation.

1. PEARSON PRODUCT MOMENT CORRELATION COEFFICIENT
The Pearson Product Moment Correlation Coefficient, denoted by 𝑟, measures the strength of the linear
relationship. To find the 𝑟, the following formula is used:
𝒓 =
𝒏(∑𝒙𝒚) − (∑ 𝒙)(∑ 𝒚)
√[𝒏(∑ 𝒙𝟐) − (∑ 𝒙)𝟐][𝒏(∑𝒚𝟐) − (∑ 𝒙)𝟐]
𝒏 = 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒑𝒂𝒊𝒓𝒆𝒅 𝒗𝒂𝒍𝒖𝒆𝒔
∑ 𝒙 = 𝒔𝒖𝒎 𝒐𝒇 𝒙 − 𝒗𝒂𝒍𝒖𝒆𝒔
∑ 𝒚 = 𝒔𝒖𝒎 𝒐𝒇 𝒚 − 𝒗𝒂𝒍𝒖𝒆𝒔
∑ 𝒙𝒚 = 𝒔𝒖𝒎 𝒐𝒇 𝒕𝒉𝒆 𝒑𝒓𝒐𝒅𝒖𝒄𝒕𝒔 𝒐𝒇 𝒑𝒂𝒊𝒓𝒆𝒅 𝒗𝒂𝒍𝒖𝒆𝒔 𝒙 𝒂𝒏𝒅 𝒚
∑ 𝒙𝟐
= 𝒔𝒖𝒎 𝒐𝒇 𝒔𝒒𝒖𝒂𝒓𝒆𝒅 𝒙 − 𝒗𝒂𝒍𝒖𝒆𝒔
∑ 𝒚𝟐
= 𝒔𝒖𝒎 𝒐𝒇 𝒔𝒒𝒖𝒂𝒓𝒆𝒅 𝒚 − 𝒗𝒂𝒍𝒖𝒆𝒔
The following table below is the interpretation of 𝑟 and can be used in interpreting the degree of linear
relationship existing between the two variables.
Value of 𝒓 Strength of Correlation
+ 1.00 Perfect Positive Correlation
+ 0.71 to +0.99 Strong Positive Correlation
+ 0.51 to +0.70 Moderately Positive Correlation
+0.31 to +0.50 Weak Positive Correlation
+0.01 to +0.30 Negligible Positive Correlation
0 No Correlation
-0.01 to -0.30 Negligible Negative Correlation
-0.31 to -0.50 Weak Negative Correlation
-0.51 to -0.70 Moderately Negative Correlation
-0.71 to -0.99 Strong Negative Correlation
-1.00 Perfect Negative Correlation
EXAMPLE 1: The table below shows the time in hours per nigh studying (x) of six grade 11 students and
their scores on a statistics test (y). Solve for the Pearson Product Correlation Coefficient, 𝑟, and interpret
the result.

2. x 1 2 3 4 5 6
y 5 10 15 15 25 35
SOLUTION:
𝒙 𝒚 𝒙𝒚 𝒙𝟐
𝒚𝟐
1 5 5 1 25
2 10 20 4 100
3 15 45 9 225
4 15 60 16 225
5 25 125 25 625
6 35 210 36 1,225
∑ 𝒙 = 𝟐𝟏 ∑ 𝒚 = 𝟏𝟎𝟓 ∑ 𝒙𝒚 = 𝟒𝟔𝟓 ∑ 𝒙𝟐
= 𝟗𝟏 ∑ 𝒚𝟐
= 𝟐, 𝟒𝟐𝟓
𝒓 =
𝒏(∑𝒙𝒚) − (∑ 𝒙)(∑ 𝒚)
√[𝒏(∑ 𝒙𝟐) − (∑ 𝒙)𝟐][𝒏(∑𝒚𝟐) − (∑ 𝒙)𝟐]
𝒓 =
𝟔(𝟒𝟔𝟓) − (𝟐𝟏)(𝟏𝟎𝟓)
√[𝟔(𝟗𝟏) − (𝟐𝟏)𝟐][𝟔(𝟐, 𝟒𝟐𝟓) − (𝟏𝟎𝟓)𝟐]
=
𝟐, 𝟕𝟗𝟎 − 𝟐, 𝟐𝟎𝟓
√[𝟓𝟒𝟔 − 𝟒𝟒𝟏][𝟏𝟒, 𝟓𝟓𝟎 − 𝟏𝟏, 𝟎𝟐𝟓
𝒓 =
𝟓𝟖𝟓
√𝟑𝟕𝟎, 𝟏𝟐𝟓
= 𝟎. 𝟗𝟔𝟏𝟓𝟕 𝒐𝒓 𝟎. 𝟗𝟔𝟐
INTERPRETATION: The value 𝑟 = 0.962 is between +0.71 to +0.99 in the table for interpretation of 𝑟. It
indicates that there is a strong positive correlation between the time in hours spent in studying and the
scores test in statistics.