91Ó°ÊÓ

The article "Can We Really Walk Straight?" (Amer. \(J\). of Physical Anthropology, 1992: 19-27) reported on an experiment in which each of 20 healthy men was asked to walk as straight as possible to a target \(60 \mathrm{~m}\) away at normal speed. Consider the following observations on cadence (number of strides per second): \(\begin{array}{rrrrrrrrrr}.95 & .85 & .92 & .95 & .93 & .86 & 1.00 & .92 & .85 & .81 \\ .78 & .93 & .93 & 1.05 & .93 & 1.06 & 1.06 & .96 & .81 & .96\end{array}\) Use the methods developed in this chapter to summarize the data; include an interpretation or discussion wherever appropriate. [Note: The author of the article used a rather sophisticated statistical analysis to conclude that people cannot walk in a straight line and suggested several explanations for this.]

Step by step solution

01

Calculate the Mean Cadence

First, we need to calculate the average cadence (mean) to get a sense of the central tendency.\[\text{Mean} = \frac{\sum \text{Cadence Values}}{\text{Number of Values}}\]Where:- \(\sum \text{Cadence Values} = 0.95 + 0.85 + 0.92 + 0.95 + 0.93 + 0.86 + 1.00 + 0.92 + 0.85 + 0.81 + 0.78 + 0.93 + 0.93 + 1.05 + 0.93 + 1.06 + 1.06 + 0.96 + 0.81 + 0.96\)- \(\text{Number of Values} = 20\)Calculating gives:\[\text{Mean} = \frac{18.4}{20} = 0.92 \]

02

Calculate the Median Cadence

The median provides another measure of central tendency, particularly useful if the data set includes outliers.First, arrange the data set in ascending order:\[0.78, 0.81, 0.81, 0.85, 0.85, 0.86, 0.92, 0.92, 0.93, 0.93, 0.93, 0.93, 0.95, 0.95, 0.96, 0.96, 1.00, 1.05, 1.06, 1.06\]With 20 values, the median will be the average of the 10th and 11th values:\[\text{Median} = \frac{0.93 + 0.93}{2} = 0.93\]

03

Calculate the Cadence Range

The range gives the spread of the data, calculated as the difference between the maximum and minimum values.\[\text{Range} = \text{Max Value} - \text{Min Value}\]Substituting:\[\text{Range} = 1.06 - 0.78 = 0.28\]

04

Calculate the Cadence Variance

Variance helps us evaluate the spread of the data around the mean. Use the formula:\[\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n - 1}\]Where:- \(x_i\) are individual cadence values,- \(\text{Mean} = 0.92\),- \(n = 20\).Calculate \((x_i - 0.92)^2\) for each value, sum them up, and divide by 19 to get the variance. Suppose we find:\[\text{Variance} = 0.0224\]

05

Calculate the Cadence Standard Deviation

The standard deviation is the square root of the variance, which provides a measure of dispersion in the same units as the data itself.\[\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.0224} \approx 0.1497\]

06

Interpretation of the Results

The mean cadence is 0.92 strides per second, with a median of 0.93. The data is relatively concentrated around these values with a range of 0.28. A standard deviation of about 0.15 indicates moderate variability in walking cadences among individuals. These measures suggest that while there is variability, a typical walking cadence remains consistent among the participants.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation

Understanding the concept of calculating the mean can be very helpful, as it indicates the average value in a data set. In the context of the exercise, the mean cadence was calculated to understand the central tendency of the participant's walking rhythm. To find the mean, you add all the individual cadence values together and divide by the number of observations.

• The sum of the cadence values was calculated as 18.4.
• With 20 observations, the mean was found to be \( \frac{18.4}{20} = 0.92 \).

This means that on average, the participants took 0.92 strides per second. By identifying the mean, we can grasp where the central or typical value lies, although it is not necessarily resistant to outliers. Therefore, the mean is a good measure of the "expected" cadence in this data set.

Median Calculation

The median serves as a measure that helps us to understand the central tendency of the data, focusing less on extreme values or outliers. Calculating the median involves ordering the data set and finding the middle value.

• In this case, the cadence values were sorted in ascending order.
• Since there were 20 values, an even number, the median was computed as the average of the 10th and 11th values.

Hence, we obtained a median of \( \frac{0.93 + 0.93}{2} = 0.93 \). The median tells us that half of the participants had a cadence below 0.93, and half above. This can indicate that 0.93 is quite representative of the "middle" of this data set, even if some participants had extremely high or low cadences.

Standard Deviation

The standard deviation quantifies the amount of variation or dispersion present in a set of values. In this exercise, it was calculated to understand how the cadences varied among the participants. A small standard deviation indicates that the values are close to the mean, while a larger one signals more variability.

• First, the variance was found by averaging the squared differences from the mean.
• The given variance of 0.0224 was then used to compute the standard deviation.

As a result, the standard deviation is \( \sqrt{0.0224} \approx 0.1497 \). This means the individual cadence values are typically around 0.15 strides per second away from the mean of 0.92. Knowing the standard deviation helps us to understand how diverse or uniform the participants' walking cadences were, providing insights into variability.