{
  "app": "mappy",
  "version": 6,
  "title": "Sample mean · formula, meaning, example",
  "savedAt": "2026-10-19T12:00:00.000Z",
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      {
        "id": "mean",
        "parentId": "statistics",
        "label": "Sample mean\n$\\bar{x}=\\frac{1}{n}\\sum_{i=1}^{n}x_i$",
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          "content": [
            {
              "type": "heading",
              "attrs": {
                "level": 2
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              "content": [
                {
                  "type": "text",
                  "text": "Sample mean: meaning before memorizing"
                }
              ]
            },
            {
              "type": "blockMath",
              "attrs": {
                "latex": "\\bar{x}=\\frac{1}{n}\\sum_{i=1}^{n}x_i"
              }
            },
            {
              "type": "paragraph",
              "content": [
                {
                  "type": "text",
                  "text": "xᵢ is observation i. n is the number of observations, with n > 0. x̄ is the arithmetic mean. Use comparable measurements in the same units."
                }
              ]
            },
            {
              "type": "heading",
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              "content": [
                {
                  "type": "text",
                  "text": "Worked example"
                }
              ]
            },
            {
              "type": "paragraph",
              "content": [
                {
                  "type": "text",
                  "text": "Hypothetical waiting times: 2, 4, and 9 minutes."
                }
              ]
            },
            {
              "type": "blockMath",
              "attrs": {
                "latex": "\\bar{x}=\\frac{2+4+9}{3}=5\\;\\text{minutes}"
              }
            },
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              "type": "heading",
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                "level": 2
              },
              "content": [
                {
                  "type": "text",
                  "text": "Change one observation"
                }
              ]
            },
            {
              "type": "blockMath",
              "attrs": {
                "latex": "\\frac{2+4+90}{3}=32\\;\\text{minutes}"
              }
            },
            {
              "type": "paragraph",
              "content": [
                {
                  "type": "text",
                  "text": "A large observation pulls the mean upward. In both examples the median is 4 minutes. Mappy renders these equations; it does not calculate them."
                }
              ]
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                      "type": "paragraph",
                      "content": [
                        {
                          "type": "text",
                          "text": "Explain why the mean changes but the median does not."
                        }
                      ]
                    }
                  ]
                }
              ]
            }
          ]
        }
      },
      {
        "id": "symbols",
        "parentId": "mean",
        "label": "Define every symbol",
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        "label": "2, 4, 9 minutes\nMean = 5 minutes",
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      {
        "id": "median",
        "parentId": "statistics",
        "label": "Median",
        "color": "#f19a48",
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          "content": [
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              "content": [
                {
                  "type": "text",
                  "text": "Median in this example"
                }
              ]
            },
            {
              "type": "paragraph",
              "content": [
                {
                  "type": "text",
                  "text": "Sort 2, 4, 9. The middle value is 4 minutes. Replacing 9 with 90 changes the mean but not the median."
                }
              ]
            }
          ]
        }
      },
      {
        "id": "compare",
        "parentId": "median",
        "label": "Middle value = 4\nCompare with the mean",
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