🔢 MathGrade 6Lesson 31
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Two-Step Data Problems

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What you will learn

  • 1Break a data question into a clear first step and a clear second step
  • 2Compute a summary statistic in step one before answering the real question in step two
  • 3Solve two-step problems involving mean, median, mode, range, and totals
  • 4Explain how the result of step one feeds into step two
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Mini Lesson

A two-step data problem is exactly what it sounds like: you cannot answer the question with one calculation. You need to do something to the data first (Step 1), and only then can you answer what was actually asked (Step 2). Most real questions about data work this way. "How far is 11 from the average?" — first you have to FIND the average, then you can subtract.

The 2-step pattern

  1. Step 1 — Organize or summarize the data. Sort it, total it, or compute a measure (mean, median, mode, range).
  2. Step 2 — Use the result of Step 1 to answer the real question (a difference, a comparison, a count, a missing value).

Worked example — distance from the mean

  • Club quiz scores: 5, 7, 8, 9, 11. Question: how far above the mean is the 11?
  • Step 1 — Find the mean. Sum = 5 + 7 + 8 + 9 + 11 = 40. Count = 5. Mean = 40 / 5 = 8.
  • Step 2 — Subtract the mean from 11. Distance = 11 - 8 = 3.
  • Answer: 11 is 3 above the mean.

Worked example — finding a missing value

  • Four scores so far: 6, 8, 9, ?. The mean of all four is 8. Find the missing score.
  • Step 1 — Use the mean to get the total. Sum = mean x count = 8 x 4 = 32.
  • Step 2 — Subtract what you have. Missing = 32 - (6 + 8 + 9) = 32 - 23 = 9.
  • Answer: the missing score is 9.

Common Step 1 actions

  • Find the median -> Step 1: sort the data.
  • Find the mean -> Step 1: add all values to get the sum.
  • Find the range -> Step 1: identify the min and the max.
  • Find the mode -> Step 1: count how many times each value appears.
  • Find a missing value (given the mean) -> Step 1: multiply mean x count to get the required sum.

Words to know

two-step problem

A problem that needs one calculation to set up the data and a second calculation to answer the question.

step 1

The setup step — sort the data, total it, or compute a measure (mean, median, mode, range).

step 2

The answer step — use the Step 1 result to find a difference, a missing value, or a comparison.

distance from mean

How far a value is above or below the mean; computed as value - mean.

missing value

A value not given directly; usually found by working backwards from a known mean or total.

The Two-step PatternA flow chart: DATA arrow STEP 1 (organize or summarize) arrow STEP 2 (answer the question). Below, lists of typical Step 1 actions (sort, add, count, find min/max, mean x count) and Step 2 actions (subtract, compare, find missing, count above).The Two-step PatternOrganize the data first. Then answer the question.DATAraw listof valuesSTEP 1organizeor summarizeSTEP 2answerquestionStep 1 might be:sort . add . count . findmin/max . mean x countStep 2 might be:subtract . compare . findmissing . count aboveStep 1 sets up the data. Step 2 uses it.
Every two-step problem follows this pattern.
Worked Example: Distance from the MeanA worked example: the data set 5, 7, 8, 9, 11. Step 1 computes the mean (sum 40 divided by 5 equals 8). Step 2 finds the distance of 11 from the mean: 11 minus 8 equals 3.Worked Example: Distance from the MeanData: 5, 7, 8, 9, 11. How far is 11 above the mean?DATA:5, 7, 8, 9, 11STEP 1find meanSum = 5 + 7 + 8 + 9 + 11 = 40Mean = 40 / 5 = 8(organize the data into one number)STEP 2subtractDistance = 11 - 8 = 311 is 3 above the mean.Step 1 result feeds Step 2. Always.
Find the mean first, then the distance.
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Tip: write "Step 1:" and "Step 2:" on your page before you start. Naming the steps stops you from mixing them up — and makes it obvious if you skipped Step 1 entirely.

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