Z score Method: Normalized score calculations

Score Normalization Results (Z-Score Method)

This report details the normalization of scores for 32 candidates across two shifts. The Z-score method was applied to ensure fairness across different difficulty levels.
Total candidates: 32
Shift1 score: 55.4, 58.6, 60.0, 62.0, 65.0, 63.0, 70.0, 71.2, 71.4, 70.4, 75.2, 70.6, 76.3, 78.3, 80.1, 81.2.
Shift2 score: 45.0, 46.0, 50.0, 51.0, 51.2, 52.5, 55.1, 55.8, 55.4, 58.3, 59.5, 60.9, 62.3, 64.2, 66.3,70.2.
Score out of 100

Step 1: The Basic Z-Score Formula

First, we find out how many "points of distance" a score is from its own shift's average. This is the Z-score.

Z = (X - μ) / σ

• X: Your raw score (e.g., 70).
• μ (Mu): The average (mean) of your specific shift.
• σ (Sigma): The standard deviation (spread) of your specific shift.

Why this works: It converts a raw mark into a "standard" number. A Z-score of 0 means you are exactly average. A Z-score of +1.5 means you performed better than about 93% of your shift.

Step 2: The Normalization Formula

Since Shift 2 was harder, we cannot just compare Z-scores directly to raw marks. We must map everyone onto a single Global Scale so they can be ranked together.

Final Score = μ_(global) + [Z × σ_(global)]

• μ_(global): The average of all 32 candidates (62.89).
• σ_(global): The standard deviation of all 32 candidates (9.72).

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Example Calculation

Let's look at a candidate from Shift 2 who scored 70:

1. Find Z-score: (70 - 56.48) / 6.94 = 1.95
   (This student is 1.95 standard deviations above their shift mean).
2. Find Final Score: 62.89 + (1.95 × 9.72) = 81.844

By using this formula, the student who got 70 in the hard shift (Shift 2) is recognized as having performed roughly as well as the student who got 85 in the easy shift (Shift 1).

Statistical Summary

Parameter	Shift 1	Shift 2	Combined
Mean (Average)	69.29	56.48	62.89
Standard Deviation	7.67	6.94	9.72

Shift 1: Normalized Scores

Candidate	Raw Score	Final Normalized Score
1	55.4	45.27
2	58.6	49.32
3	60.0	51.10
4	62.0	53.64
5	65.0	57.44
6	63.0	54.91
7	70.0	63.78
8	71.2	65.31
9	71.4	65.56
10	70.4	64.29
11	75.2	70.38
12	70.6	64.54
13	76.3	71.77
14	78.3	74.31
15	80.1	76.59
16	81.2	77.99

Shift 2: Normalized Scores

Candidate	Raw Score	Final Normalized Score
1	45.0	46.81
2	46.0	48.21
3	50.0	53.81
4	51.0	55.21
5	51.2	55.49
6	52.5	57.31
7	55.1	60.95
8	55.8	61.93
9	55.4	61.37
10	58.3	65.43
11	59.5	67.12
12	60.9	69.08
13	62.3	71.04
14	64.2	73.70
15	66.3	76.64
16	70.2	82.10