Visual psychophysicists have recently developed tools to measure the maximal speed at which the brain can accurately carry out different types of computations (H. Kirchner & S. J. Thorpe, 2006). We use this methodology to measure the maximal speed with which individuals can make magnitude comparisons between two single-digit numbers. We find that individuals make such comparisons with high accuracy in 306 ms on average and are able to perform above chance in as little as 230 ms. We also find that maximal speeds are similar for “larger than” and “smaller than” number comparisons and in a control task that simply requires subjects to identify the number in a number–letter pair. The results suggest that the brain contains dedicated processes involved in implementing basic number comparisons that can be deployed in parallel with processes involved in low-level visual processing.

*distance effect*(Dehaene, 1997; Dehaene & Changeux, 1993) in binary numerical comparisons, which states that accuracy increases and response times decrease with the absolute distance between the two numbers being compared. This effect has been replicated in a number of studies (Dehaene, 1989, 1996) and has motivated the “mental number line hypothesis,” which states that numeric magnitude is spatially encoded in the brain (Dehaene, Bossini, & Giraux, 1993; Moyer & Landauer, 1967). Song and Nakayama (2008) investigated this hypothesis using a visually guided manual reaching task in which subjects were asked to compare a single digit number displayed at the center of the screen with the number “5” by reaching with their index finger toward the appropriate location on the screen (left, center, or right of the displayed number to signify smaller, equal, or larger, respectively). They found that the greater the numeric deviation from 5, the greater the deviation of the pointing trajectory.

*smaller*of the two digits under speed instructions in order to investigate if it is equally easy for the brain to make rapid “larger than” and “smaller than” number comparisons. Finally, in Experiment 3, we compare these results to a number identification control task in which subjects are shown a letter–number pair and asked to identify the location of the number.

*X*

_{ i }denote the accuracy of the

*i*th ordered response (1 = correct, 0 = incorrect). An exponentially weighted moving average (EWMA) measure of accuracy is then computed using the following formula:

*λ*is a parameter indicating how much weight to give to past (ordered) observations in the moving average. Note that when

*λ*= 1, the EWMA statistic is based only on the most recent observation. In contrast, as

*λ*approaches 0 previous observations are given increasing weight relative to the latest observation. EWMA

_{0}was set to 0.5 (i.e., chance performance).

*i*observations given by

*μ*=

*E*(

*X*

_{ i }) = 0.5. The second term provides an expression for the width of the confidence interval:

*N*is the number of standard deviations included in the confidence interval, and

*σ*= Std(

*X*

_{ i }) = 0.5 is the standard deviation of each observation

*X*

_{ i }under the null hypothesis. Note that under the null hypothesis performance in every trial depends on the independent flip of a fair coin.

*λ*= 0.01 and

*N*= 3. The EWMA analysis was run separately for each subject and condition.

*SEM*) for the speed and 345 ± 15 ms for the accuracy condition. The average per-subject difference in reaction times between the two conditions (accuracy–speed) was 39 ± 11 ms (two-sided paired

*t*-test,

*p*= 0.0036). Average accuracies were 91.2 ± 1.0% in the speed condition and 95.4 ± 1.0% in the accuracy condition. The average per-subject accuracy difference between the two conditions (accuracy–speed) was 4.3 ± 1.1% (two-sided paired

*t*-test,

*p*= 0.0030). As expected, subjects were faster and less accurate in the speed condition.

S | N | Accuracy (%) | Mean RT (ms) ± SEM | MRT |
---|---|---|---|---|

Experiment 1: Speed condition | ||||

1* | 882 | 86.8 | 409 ± 2.16 | 237 |

2* | 897 | 91.8 | 310 ± 1.74 | 248 |

3* | 896 | 90.7 | 265 ± 1.51 | 200 |

4* | 886 | 94.7 | 277 ± 1.72 | 200 |

5* | 898 | 89.4 | 272 ± 1.58 | 164 |

6* | 891 | 90.7 | 261 ± 1.39 | 205 |

7 | 878 | 86.4 | 285 ± 2.65 | 220 |

8 | 889 | 97.6 | 401 ± 2.42 | 303 |

9 | 896 | 93.8 | 297 ± 1.19 | 244 |

10 | 887 | 94.5 | 295 ± 1.38 | 239 |

11 | 894 | 86.8 | 268 ± 1.26 | 221 |

12 | 898 | 90.5 | 332 ± 1.39 | 273 |

All | 10692 | 91.2 ± 1.0 | 306 ± 15 | 230 ± 11 |

Experiment 1: Accuracy condition | ||||

1* | 897 | 97.4 | 405 ± 2.30 | 310 |

2* | 900 | 97.9 | 366 ± 2.50 | 244 |

3* | 895 | 97.9 | 311 ± 2.35 | 235 |

4* | 891 | 96.7 | 292 ± 1.90 | 221 |

5* | 893 | 95.5 | 290 ± 1.50 | 220 |

6* | 900 | 95.3 | 271 ± 1.53 | 202 |

7 | 870 | 97.0 | 420 ± 3.11 | 209 |

8 | 893 | 97.2 | 413 ± 3.16 | 312 |

9 | 897 | 96.0 | 317 ± 1.57 | 255 |

10 | 896 | 97.5 | 357 ± 2.32 | 258 |

11 | 892 | 86.7 | 318 ± 1.82 | 247 |

12 | 865 | 90.1 | 385 ± 3.21 | 290 |

All | 10689 | 95.4 ± 1.0 | 345 ± 15 | 250 ± 11 |

*t*-test,

*p*= 0.014). For better comparison with the previous literature, we also computed the binned MRT measure of KT 2006, which is computed using the pooled data. We chose 10-ms bins centered at 10 ms, 20 ms, etc., and a threshold of 10 consecutive bins at a 5% confidence level. The resulting MRT statistic was 170 ms (i.e., the 165- to 175-ms bin) for the speed task and 190 ms for the accuracy task. This result shows that our MRT measures are more conservative than the KT measures and give more confidence that the average subject is indeed performing at above-chance levels at the reported MRT values.

*t*-test,

*p*= 0.0021) in the speed task and −5.6 ± 0.7 ms per unit (two-sided

*t*-test,

*p*= 5.9 × 10

^{−6}) in the accuracy task. The mean fitted accuracy slope was 0.53 ± 0.08 (two-sided

*t*-test,

*p*= 2.6 × 10

^{−5}) in the speed task and 0.60 ± 0.12 (two-sided

*t*-test,

*p*= 4.0 × 10

^{−4}) in the accuracy task. Thus, in both cases, responses became faster and more accurate as a function of numerical distance, confirming the presence of a numerical distance effect in our experimental setup.

S | RT constant (ms/unit) ± SEM | RT slope (ms/unit) ± SEM | Accuracy constant ± SEM | Accuracy slope ± SEM |
---|---|---|---|---|

Experiment 1: Speed condition | ||||

1* | 414 ± 18 | −1.3 ± 1.0 | 0.82 ± 0.03 | 0.340 ± 0.003 |

2* | 321 ± 11 | −3.3 ± 0.6 | 0.59 ± 0.06 | 0.69 ± 0.01 |

3* | 275 ± 8.4 | −2.8 ± 0.5 | 0.52 ± 0.05 | 0.659 ± 0.009 |

4* | 289 ± 11 | −3.3 ± 0.6 | 0.93 ± 0.09 | 0.79 ± 0.02 |

5* | 271 ± 9.3 | 0.3 ± 0.5 | 0.85 ± 0.04 | 0.435 ± 0.005 |

6* | 273 ± 7.1 | −3.2 ± 0.4 | 0.81 ± 0.05 | 0.514 ± 0.007 |

7 | 295 ± 26 | −2.7 ± 1.4 | 0.97 ± 0.03 | 0.273 ± 0.003 |

8 | 441 ± 20 | −10.9 ± 1.1 | 1.8 ± 0.2 | 0.76 ± 0.04 |

9 | 310 ± 5.1 | −3.34 ± 0.3 | 0.88 ± 0.07 | 0.72 ± 0.02 |

10 | 306 ± 6.9 | −3.0 ± 0.4 | 0.79 ± 0.09 | 0.84 ± 0.02 |

11 | 274 ± 5.9 | −1.5 ± 0.3 | 0.60 ± 0.03 | 0.428 ± 0.004 |

12 | 339 ± 7.1 | −2.0 ± 0.4 | 2.55 ± 0.05 | −0.076 ± 0.003 |

All | 317 ± 16 | −3.1 ± 0.8 | 1.0 ± 0.2 | 0.53 ± 0.08 |

Experiment 1: Accuracy condition | ||||

1* | 442 ± 18 | −10.0 ± 1.0 | 1.6 ± 0.2 | 0.85 ± 0.05 |

2* | 382 ± 23 | −4.1 ± 1.3 | 2.3 ± 0.2 | 0.57 ± 0.03 |

3* | 330 ± 20 | −5.3 ± 1.1 | 1.4 ± 0.25 | 1.12 ± 0.09 |

4* | 311 ± 13 | −5.2 ± 0.7 | 1.8 ± 0.1 | 0.61 ± 0.02 |

5* | 306 ± 8.1 | −4.2 ± 0.4 | 1.82 ± 0.09 | 0.43 ± 0.01 |

6* | 286 ± 8.5 | −4.0 ± 0.5 | 0.4 ± 0.1 | 1.24 ± 0.05 |

7 | 449 ± 35 | −8.1 ± 1.9 | 2.4 ± 0.1 | 0.36 ± 0.02 |

8 | 450 ± 35 | −10.1 ± 1.9 | 2.3 ± 0.1 | 0.46 ± 0.02 |

9 | 331 ± 9.0 | −3.6 ± 0.5 | 0.9 ± 0.1 | 0.98 ± 0.04 |

10 | 374 ± 20 | −4.7 ± 1.1 | 1.9 ± 0.2 | 0.70 ± 0.04 |

11 | 331 ± 12 | −3.7 ± 0.7 | 1.58 ± 0.03 | 0.084 ± 0.002 |

12 | 401 ± 39 | −4.3 ± 2.1 | 2.99 ± 0.06 | −0.192 ± 0.002 |

All | 366 ± 17 | −5.6 ± 0.7 | 1.8 ± 0.2 | 0.6 ± 0.1 |

*p*= 0.67 and RT = −14 ± 30 ms,

*p*= 0.65) or the accuracy condition results (speed condition first: accuracy = 96.8 ± 0.5% and RT = 323 ± 21 ms; accuracy condition first: accuracy = 94.1 ± 1.9% and RT = 368 ± 18 ms; difference: accuracy = 2.7 ± 1.9%,

*p*= 0.19 and RT = −46 ± 28 ms,

*p*= 0.13).

*smaller*number. The goal was to investigate if the brain could execute “larger than” and “smaller than” number comparisons at similar maximal speeds.

*t*-test,

*p*= 0.38). The mean accuracy for the task across all subjects was 84.6 ± 1.9%. The difference in mean accuracy was statistically significant (Experiment 2 − Experiment 1 speed: −6.6 ± 2.2%; two-sided

*t*-test,

*p*= 0.0060).

Experiment 2: “Smaller than” comparisons | ||||
---|---|---|---|---|

S | N | Accuracy (%) | Mean RT (ms) ± SEM | MRT |

13 | 900 | 86.2 | 301 ± 1.4 | 213 |

14 | 885 | 85.3 | 334 ± 3.0 | 184 |

15 | 893 | 86.7 | 389 ± 2.5 | 295 |

16 | 847 | 70.7 | 289 ± 4.2 | 275 |

17 | 899 | 91.7 | 336 ± 2.1 | 225 |

18 | 898 | 87.6 | 314 ± 1.9 | 259 |

19 | 899 | 95.1 | 314 ± 1.7 | 250 |

20 | 896 | 83.7 | 315 ± 1.9 | 250 |

21 | 883 | 86.5 | 296 ± 2.7 | 203 |

22 | 898 | 77.4 | 282 ± 1.9 | 242 |

23 | 893 | 77.5 | 378 ± 2.9 | 309 |

24 | 900 | 86.4 | 310 ± 1.7 | 259 |

All | 5344 | 84.6 ± 1.9 | 322 ± 10 | 247 ± 11 |

*t*-test,

*p*= 0.255).

*t*-test; RT:

*p*= 0.61, accuracy:

*p*= 0.65), even though at each numerical distance the estimated accuracy was higher in the “larger than” condition than in the “smaller than” condition. There was a significant difference in the slope of the accuracy curve (Experiment 2 − Experiment 1 speed: −0.25 ± 0.08; two-sided

*t*-test,

*p*= 0.0064), though not in the slope of the RT curve (Experiment 2 − Experiment 1 speed: 1.66 ± 0.89 ms; two-sided

*t*-test,

*p*= 0.08).

Experiment 2: “Smaller than” comparisons | ||||
---|---|---|---|---|

S | RT constant (ms/unit) ± SEM | RT slope (ms/unit) ± SEM | Accuracy constant ± SEM | Accuracy slope ± SEM |

13 | 302 ± 7.8 | −0.4 ± 0.4 | 0.58 ± 0.03 | 0.417 ± 0.004 |

14 | 334 ± 32 | 0.06 ± 1.7 | 0.89 ± 0.03 | 0.269 ± 0.003 |

15 | 398 ± 23 | −2.2 ± 1.3 | 1.43 ± 0.03 | 0.127 ± 0.002 |

16 | 298 ± 67 | −2.2 ± 3.6 | 0.28 ± 0.02 | 0.174 ± 0.001 |

17 | 330 ± 17 | 1.6 ± 0.9 | 0.94 ± 0.05 | 0.518 ± 0.008 |

18 | 322 ± 13 | −2.1 ± 0.7 | 0.78 ± 0.04 | 0.387 ± 0.004 |

19 | 329 ± 11 | −3.9 ± 0.6 | 2.00 ± 0.08 | 0.317 ± 0.008 |

20 | 326 ± 14 | −3.0 ± 0.8 | 0.94 ± 0.03 | 0.209 ± 0.002 |

21 | 299 ± 27 | −0.7 ± 1.5 | 0.98 ± 0.03 | 0.275 ± 0.003 |

22 | 292 ± 14 | −2.6 ± 0.7 | 0.34 ± 0.02 | 0.268 ± 0.002 |

23 | 379 ± 33 | −0.3 ± 1.8 | 0.52 ± 0.02 | 0.210 ± 0.002 |

24 | 316 ± 10 | −1.5 ± 0.6 | 1.28 ± 0.03 | 0.170 ± 0.002 |

All | 327 ± 9.3 | −1.4 ± 0.4 | 0.9 ± 0.1 | 0.28 ± 0.03 |

*p*= 0.92) but a significant difference between the two conditions (

*p*= 0.0029) with an estimated set size of 17.7 ms with no interaction between the two (

*p*= 0.99).

*t*-test,

*p*= 0.0096). The mean accuracy for the task across subjects was 89.1 ± 1.8%, which was not significantly different from the number comparison task (Experiment 3 − Experiment 1 speed: −2.1 ± 2.1%; two-sided

*t*-test,

*p*= 0.33).

Experiment 3: Number identification control | ||||
---|---|---|---|---|

S | N | Accuracy (%) | Mean RT (ms) ± SEM | MRT |

25 | 600 | 92.0 | 260 ± 1.8 | 204 |

26 | 600 | 96.0 | 292 ± 1.8 | 228 |

27 | 600 | 94.0 | 260 ± 1.9 | 185 |

28 | 576 | 90.5 | 288 ± 2.7 | 220 |

29 | 598 | 89.0 | 256 ± 2.8 | 188 |

30 | 599 | 93.5 | 271 ± 2.1 | 226 |

31 | 590 | 92.9 | 270 ± 2.2 | 215 |

32 | 591 | 94.8 | 251 ± 2.1 | 185 |

33 | 598 | 84.3 | 268 ± 2.7 | 217 |

34 | 593 | 85.8 | 253 ± 1.9 | 212 |

35 | 592 | 76.9 | 231 ± 3 | 212 |

36 | 590 | 79.7 | 249 ± 3.7 | 232 |

All | 3564 | 89.1 ± 1.8 | 262 ± 5 | 210 ± 5 |

*t*-test,

*p*= 0.11). In other words, there is no evidence that comparing two numbers and choosing the larger one takes longer than identifying a number over a letter.