Scenario: A local community centre runs a health awareness campaign about hydration. Nine participants recorded how many cups of water they drank per day during a typical weekday. The results (in cups) are: 5, 6, 7, 6, 8, 10, 7, 9, 6Task: a) Calculate the mean, variance, and standard deviation (to two decimals) for this sample. Include all steps, including the sum of squared deviations: ∑(x − x̄)². b) Interpret the results in the context of daily hydration habits. What do the measures of central tendency and spread tell you about how the group behaves? Clearly explain what the numerical value of the standard deviation means in this context, and suggest a realistic goal based on your interpretation. c) Another group recorded an average of 7.00 cups per day with a standard deviation of 0.71 cups.Compare the two groups’ data and explain which group is more consistent. Then, describe how this finding could help a community leader or coach make a realistic recommendation to participants.

Question

BlackTom AI · Accepted Answer

a) Calculations:
Given data: 5, 6, 7, 6, 8, 10, 7, 9, 6

The mean amount of water drunk is 64/9 = 7.11 cups.

The mean of these values is found by working out the sum of them all and then dividing by the number of entries, i.e. 5. So (5 + 6 + 7 + 6 + 8 + 10 + 7 + 9 + 6)/9 = 5.55. Now the variance of the set is the sum of the squared differences between each value and the mean divided by the number of entries in the set.

Thus, the variance = [(5-5.55)² + (6-5.55)² + (7-5.55)² + (6-5.55)² + (8-5.55)² + (10-5.55)² + (7-5.55)² + (9-5.55)² + (6-5.55)²]/9 = 20.89.

Sample variance: s² = ∑(x − x̄)² / (n-1) = 20.89 / 8 ≈ 2.61

Standard deviation = 2.61 (2.61)^0.5 = 1.62 cups

Summary:
- Mean: 7.11 cups
There existed a variance of 2.61.
- Standard deviation: 1.62 cups
- Deviations from the mean: 2.35, 4.7, 6.0, 5.0, 2.0, 3.0, 1.0, 3.0, 4.0, 2.0, 5.

b) Interpretation:
The mean daily water intake was 7.11 cups indicating that the average person drank a little over 7 cups of water each day, indicating good hydration overall. These results suggest that women in the United States consume on average 1.65 cups of green and yellow vegetables per day with a standard deviation of 1.62 cups. The results show that individuals vary widely in their water intake, with some consuming as few as 5 glasses of water per day and others drinking as many as 10. It seems sensible to target a quantity of 7 or 8 cups of water per day but given the considerable difference in the amount reported by the individuals it may be necessary to remind people to make a habit of drinking more water.

c) Comparison and Recommendation:
In this second group the mean water intake was 7.0 and the standard deviation 0.7, implying a hydration pattern which shows much less variation in the quantity of water consumed either from day to day or from one person to another. The first group of companies show less uniformity. A community leader may aim for consistency among the group, so suggesting a general goal of 'about seven cups a day for everyone' and monitoring individual consumption to identify large deviations could lead to the development of healthy routines.

MDM4U - Mathematics of Data Management 12 (2025-26) - A

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