Calculate and interpret the coefficient of determination

What is the first calculation when finding the coefficient of determination from r?

Predict how increasing scatter size changes the pattern. Compare 0, 6 and 12. The dashed line is the fixed underlying model, not a line joining observations. Both axis scales stay fixed

What does increasing this scatter size do?

Connect r² to the fitted linear model

For simple least-squares linear regression with an intercept, r² compares the fitted line with using the response mean for every case. Both variables must vary. State this model before interpreting a squared coefficient as explained response variation

1. 1State the model Here the model is simple least-squares linear regression with an intercept, fitted to paired numerical data with variation in both variables. In this setting r² is the proportion of observed response variation accounted for by the fitted line relative to the mean-only baseline
2. 2Calculate and report Square the signed r, then multiply by 100 for a percentage. Keep the full supplied value through the calculation and round the final report. The unexplained proportion is 1−r² in the same model
3. 3Keep the interpretation limited Explained variation is a statistical description of squared deviations, not a percentage of people, a percentage increase, proof of causation or a guarantee of prediction accuracy. Inspect the plot and model assumptions
4. 4Direction is separate Opposite correlations have the same square. Recovering signed r from r² requires independent direction information; a squared coefficient alone cannot select a sign

Given values

\(\displaystyle r=-0.6\)

Square the supplied correlation

Given values

\(\displaystyle r^2=0.36\)

Write the explained percentage as a number, without a percent sign

Keeping the sign in r². One of these lines is right. Which one?

For the stated simple linear model with an intercept, r=−0.8. Which interpretation is supported?

Given values

\(\displaystyle r^2=0.64\)

Find the unexplained percentage as a number, without a percent sign

A fitted association has r²=0.49, but its direction is not supplied. What can be concluded about r?

Given values

\(\displaystyle r^2=0.49\)

A separate scatterplot is described as having a falling tendency. Recover r to two decimal places

Two fitted linear models report rA=−0.8 and rB=+0.6. Which has the larger proportion of response variation explained?

Reconcile a correlation report with the fitted data

Reuse the six fictional sessions from 2F: (minutes,cards)=(1,3),(2,4),(3,6),(4,5),(5,8),(6,7). Their exact correlation is 31/35. A simple least-squares line with an intercept has been fitted: ŷ=12/5+(31/35)x. You are given the line; deriving its coefficients is a later regression task. Calculate r² and the explained/unexplained percentages to one decimal place, then write a contextual report

1. 1Check two representations The response mean is 5.5. Calculate each fitted value and residual y −ŷ. Compare the sum of squared residuals with the total sum of squares about 5.5. Verify that 1−SSE/SST agrees with the squared correlation; retain exact fractions or full precision before rounding
2. 2Choose and document technology Use a spreadsheet or suitable calculator. Keep the paired rows and a labelled XY plot. Record CORREL or RSQ with the exact same six rows; independently calculate predicted values, residuals and squared deviations. State why these functions and the chosen window suit the comparison. Critique “78.4% of sessions were caused by extra time” and “the model guarantees every prediction”
3. 3Save and review your complete response Retain the original data, chosen functions and windows, independent checks and written interpretation. Explain why the selected tools suit the task. Independent production and actual external-tool execution are not automatically marked; use self or teacher review before reading the model

Model response: two calculations agree

r²=961/1225≈0.7844898, so 78.4% is explained and 21.6% is unexplained to one decimal place. In these six fictional sessions, the fitted simple linear model accounts for about 78.4% of the observed variation in cards completed relative to using the mean card count for every session. This is a sample/model description, not a causal conclusion or a prediction guarantee

1. 1Independent residual check In row order the residuals are(−10,−6,33,−33,41,−25)/35. Their squared sum is 132/35. The total sum of squares about 5.5 is 35/2. Hence 1−(132/35)/(35/2)=1−264/1225=961/1225. This identity relies on the stated least-squares line with an intercept, not an arbitrary line
2. 2Reproducible tool record Enter minutes in B2:B7 and cards in C2:C7. Compare =CORREL(B2:B7,C2:C7)^2 and =RSQ(C2:C7,B2:B7). In D2 enter =12/5+(31/35)*B2; in E2 enter =C2-D2 and fill both down. In F2 enter =E2^2 and in G2 enter =(C2-AVERAGE($C$2:$C$7))^2, then fill both down. Sum F for SSE and G for SST; compare 1−SSE/SST. Use B horizontally and C vertically on a labelled XY plot. Record precision and retain the exact original rows
3. 3Limits and comparison The two causal/prediction claims are unsupported. The percentage is not a count of sessions, and unexplained variation does not identify a particular cause. A larger r² across different response variables compares proportions, not raw squared-error sizes, population reliability or intervention effects

Given values

\(\displaystyle r=\frac{31}{35}\)

Report the explained percentage to one decimal place, without a percent sign