Bloom's 2 sigma problem

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Bloom's 2 sigma problem refers to an educational phenomenon observed by educational psychologist Benjamin Bloom and initially reported in 1984 in the journal Educational Researcher. Bloom found that the average student tutored one-to-one using mastery learning techniques performed two standard deviations better than students who learn via conventional instructional methods—that is, "the average tutored student was above 98% of the students in the control class".[1]:4 Additionally, the variation of the students' achievement changed: "about 90% of the tutored students ... attained the level of summative achievement reached by only the highest 20%" of the control class.[1]:4 Bloom's graduate students J. Anania and A. J. Burke conducted studies of this effect at different grade levels and in different schools, observing students with "great differences in cognitive achievement, attitudes, and academic self-concept".[1]:15

Objects of change process[edit]

Though Bloom concluded that one-to-one tutoring is "too costly for most societies to bear on a large scale", Bloom conjectured that a combination of two or three altered variables may result in a similar performance improvement. Bloom thus challenged researchers and teachers to "find methods of group instruction as effective as one-to-one tutoring".[1]:15

Bloom classified alterable variables that may have, in combination, a 2 sigma effect as the following "objects of change process":

  1. Learner
  2. Instructional material
  3. Home environment or peer group
  4. Teacher

Bloom and his graduate students considered and tested various combinations of these variables, focusing only on those variables that individually had a 0.5 or higher effect size. These included:

Effect of selected alterable variables on student achievement. Adapted from,[1]:6 Walberg (1984).
Object of change process Alterable variable Effect size Percentile equivalent
Teacher Tutorial instruction 2.00 98
Teacher Reinforcement 1.2
Learner Feedback-corrective (Mastery Learning) 1.00 84
Teacher Cues and explanations 1.00
Teacher, Learner Student classroom participation 1.00
Learner Student time on task 1.00
Learner Improved reading/study skills 1.00
Home environment / peer group Cooperative learning 0.80 79
Teacher Homework (graded) 0.80
Teacher Classroom morale 0.60 73
Learner Initial cognitive prerequisites 0.60
Home environment / peer group Home environment intervention 0.50 69


Considering the significant outcomes of these studies on student performance, educational researchers can make a number of implications and conjectures for follow-up studies. Among them:

  • Labeling students as low achievers is less relevant, since altering one or two variables can have significant positive effects on the average learner.
  • Technology may simulate tutoring effects without the high cost of providing a live tutor for each student.
  • Social aspects present in one-to-one tutoring may imply a larger role for consideration of sociality in (or the social nature of) learning.


  1. ^ a b c d e Bloom, Benjamin S (June–July 1984). "The 2 Sigma Problem: The Search for Methods of Group Instruction as Effective as One-to-One Tutoring" (PDF). Educational Researcher. 13 (6): 4–16. doi:10.3102/0013189x013006004. S2CID 1714225. Retrieved 9 June 2018.

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