Week Five:

Math and Stats Fundamentals

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Why sound math skills for safety performance measurement?

- Measurements / metrics should be reliable and accurate
- Collected data must be analyzed
- Useful comparisons between results and goals
- Determine trends / changes
- Validate controls
- Validate analysis methods
- Reliable forecasting

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Data Formats

- Categorical Data
- Categories (i.e., male / female; departments, etc.)
- Ordinal Data
- Survey Data
- Likert Scales
- Interval Data
- Ratio Data

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Categorical Data

- Categories (i.e., male / female; departments, etc.)
- Only differentiate membership in a group
- Least useful from statistical analysis standpoint

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Ordinal Data

- “order” / “ordering”
- Survey Data (i.e. Likert Scales)
- No value comparisons
- More useful statistically than categorical, but low

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Interval Data

- Continuous / continuous scale
- Equality between points on the scale
- Zero is simply a “place holder”
- Fair degree of flexibility
- Example: Fahrenheit / Celsius thermometer
- More statistically useful than categorical and ordinal

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Ratio Data

- Continuous data
- Zero is not simply a placeholder (represents the absence of a characteristic)
- Magnitude between values exist
- Counting number of instances
- Highest degree of statistical usefulness

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Descriptive Statistics

- Population Data
- Measures of Central Tendency
- Mean
- Median
- Mode
- Measures of Variability
- Range
- Variance
- Standard Deviation
- Correlation Coefficient

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Inferential Statistics

- Sample Data
- Statistics that Allow for an Inference
- Sampling Distribution
- Differences between Means
- Chi Square

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Mean

Mean =

Σ X

n

Σ X = sum of the individual items / observations / values

n = total number of individual items / observations / values

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Median

- Point where 50% of the values lie above and 50% lie below
- First arrange values / items from lowest to highest
- If odd # of values / items, then median is the “middle” value / item
- If even # of values / items, then average the two “middle” values / items

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Mode

- Most Frequently Occurring #
- There may be more than one mode in a set of data

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Range

- Difference between the lowest value and the highest value in the distribution
- Arrange from lowest to highest; subtract lowest from highest

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Variance for Samples

σ² =

Σ (x-mean)² + (y-mean)²

N-1

N= total number of observation

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Variance for Total Population

σ² =

Σ (x-mean)² + (y-mean)²

N

N= total number of observation

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Standard Deviation

√σ²

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Standard Deviation =

√

Σ (v1 – mean)² + (v2-mean)²….

n

Calculate Std. Deviation

* If entire population sampled.

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Standard Deviation =

√

Σ (v1 – mean)² + (v2-mean)²….

(n -1)

Calculate Std. Deviation

* If sample of population.

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Normal Distribution

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UCL

LCL

MEAN

## Chart1

18 |

23 |

14 |

17 |

21 |

33 |

20 |

25 |

22 |

12 |

12 |

10 |

## Sheet1

J | F | M | A | M | J | J | A | S | O | N | D |

18 | 23 | 14 | 17 | 21 | 33 | 20 | 25 | 22 | 12 | 12 | 10 |

## Sheet1

## Sheet2

## Sheet3

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Creating A Control Chart: Steps

- Plot Data on A Graph
- Calculate and Place Mean on the Chart
- Calculate and Place Control Limits

UCL / LCL Calculations for #s of Events / Samples

- 95% Statistical Significance = 2 std. deviations from mean = 1.96 = normal distribution
- UCL = X + (Z x S)
- LCL = X – (Z x S)
- X = mean
- Z = normal distribution (in safety use 1.96)
- S= Std. Deviation of Population

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- 95% Statistical Significance = 2 std. deviations from mean = 1.96 = normal distribution
- UCL = p + 1.96 √ (p(1-p)/n)
- LCL = p – 1.96 √ (p(1-p)/n)
- p= mean proportion / %

UCL / LCL Calculations for Proportions / %

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Correlations

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Correlations

## Chart1

5 | 14 |

7 | 12 |

9 | 10 |

11 | 8 |

13 | 3 |

15 | 0 |

## Sheet1

Safe Behavior | Incidents | |

Qtr 1 | 5 | 14 |

Qtr 2 | 7 | 12 |

Qtr 3 | 9 | 10 |

Qtr 4 | 11 | 8 |

Qtr 5 | 13 | 3 |

Qtr 6 | 15 | 0 |

Qtr7 | 18 | 2 |

Qtr 8 | 21 | 1 |

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Correlations

## Chart1

5 | 9 |

7 | 10 |

5 | 8 |

8 | 7 |

4 | 9 |

6 | 7 |

3 | 8 |

4 | 4 |

## Sheet1

Safe Behavior | Incidents | |

Qtr 1 | 5 | 9 |

Qtr 2 | 7 | 10 |

Qtr 3 | 5 | 8 |

Qtr 4 | 8 | 7 |

Qtr 5 | 4 | 9 |

Qtr 6 | 6 | 7 |

Qtr7 | 3 | 8 |

Qtr 8 | 4 | 4 |

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Correlation: Purpose?

## Chart1

10 | 19 |

18 | 22 |

28 | 24 |

36 | 25 |

40 | 22 |

## Sheet1

Column1 | # of Obs. | Safe Behavior |

Qtr 1 | 10 | 19 |

Qtr 2 | 18 | 22 |

Qtr 3 | 28 | 24 |

Qtr 4 | 36 | 25 |

Qtr 5 | 40 | 22 |

To update the chart, enter data into this table. The data is automatically saved in the chart. |

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Pearson Correlation Coefficient

Tip: Use a Calculator with stats functions.

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Correlations – The Numbers

Strong Negative

Strong Positive

No Correlation

-1

0

+1

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N =

4 (1-p)

S² p

N= Total Number of Observations / Samples

p= % safe / % unsafe observed

S= Desired Level of Accuracy

95% Confidence Level – Two Std. Deviations from Mean

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Example:

Calculating Observation Reliability

- Behavioral Observations: Safe Forklift Operation While Traveling in Warehouse
- Observed: 75% Safe
- Desire 10% Accuracy Level
- 4 (1 -.75) / (0.01 X .75)
- # of Observations Necessary = 133 (That’s a lot if you only have a few forklift drivers!)

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Monthly Accident Control Chart

0

10

20

30

40

Month

# of Accidents

Series1

182314172133202522121210

JFMAMJJASOND