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In a uniform distribution, calculating the mean is straightforward. The range of values is uniformly spread out, meaning each interval of equal length is equally likely. To find the mean of a uniform distribution confined between two endpoints, you simply take the average of those two endpoints. For example, let's consider a situation where the shampoo volume is uniformly distributed between 374 ml and 380 ml. The formula for calculating the mean, when given endpoints \( a \) and \( b \), is:\[\text{Mean} = \frac{a + b}{2}\]Inserting the given values, \( a = 374 \) and \( b = 380 \), the calculation becomes:\[\frac{374 + 380}{2} = 377\text{ ml}\]This mean tells us that on average, the container holds 377 ml of shampoo.

Understanding the variability or spread of the data is essential, and this is captured by the standard deviation. For a continuous uniform distribution, the standard deviation is calculated using the following formula:\[\text{Standard Deviation} = \frac{b - a}{\sqrt{12}}\]Where \( b \) and \( a \) are the upper and lower bounds, respectively. In our scenario, \( a = 374 \) ml, and \( b = 380 \) ml:\[\text{Standard Deviation} = \frac{380 - 374}{\sqrt{12}} = \frac{6}{\sqrt{12}} \approx 1.732\text{ ml}\]The standard deviation here denotes how spread out the volumes are around the mean. This is crucial for understanding the variability in product filling.

Finding percentiles in a uniform distribution can give insights into the probability of an event occurring below a certain threshold. For example, determining the volume of shampoo surpassed by \(95\%\) of the containers requires calculating the 95th percentile. The formula used is:\[x = a + (b-a) \times P\]Where \( P \) is the desired percentile (in this case, \(0.95\)), and \( a \) and \( b \) are the endpoints. Applying this with \( P = 0.95\):\[x = 374 + (380 - 374) \times 0.95 = 374 + 6 \times 0.95 = 379.7\text{ ml}\]This value, 379.7 ml, indicates that \(95\%\) of shampoo volumes are less than or equal to this amount, a useful metric for quality control.

The probability of an event in a uniform distribution is essentially the length of the interval representing the event, divided by the total length of the distribution's interval. This principle allows for calculating probabilities for specific conditions, like containers filled with less than a certain volume. For instance, calculating the probability that a container holds less than 375 ml involves:\[P(X < 375) = \frac{375 - 374}{380 - 374} = \frac{1}{6} \approx 0.1667\]This result, approximately \(16.67\%\), represents the likelihood that any given container will have less than 375 ml of shampoo. Such calculations are valuable for makers to understand if the product meets specified targets.