Question

the times (in seconds) of 8 runners for the 100 - yard dash are recorded and used to determine the following calculations: mean 11.8, median 12.2, mode 12.5, range 3.2, iqr 2.4. the time - keeper had difficulties with the stopwatch and did not start the time until the runners had been running for 1.5 seconds. what would these calculations be if this 1.5 seconds were added to each runners 100 - yard dash time? • press each hotspot. • label the corresponding number below with the requested value. press to hear a reminder or hint for this problem. adding or subtracting the same value from every data value in a set does not change the measures of dispersion.

Explanation:

Step1: Effect on mean

If we add a constant \(c = 1.5\) to each data - point \(x_i\) in a set of data, the new mean \(\bar{y}\) of the new data - set \(y_i=x_i + c\) is given by \(\bar{y}=\bar{x}+c\). The original mean \(\bar{x}=11.8\), so the new mean is \(11.8 + 1.5=13.3\).

Step2: Effect on median

The median is the middle - value of a data - set. When we add a constant \(c = 1.5\) to each data - point, the position of the middle - value does not change, but its value increases by the constant. The original median is \(12.2\), so the new median is \(12.2+1.5 = 13.7\).

Step3: Effect on mode

The mode is the most frequently occurring value in a data - set. When we add a constant \(c = 1.5\) to each data - point, the most frequently occurring value also increases by the constant. The original mode is \(12.5\), so the new mode is \(12.5 + 1.5=14\).

Step4: Effect on range

The range is \(R=\text{max}-\text{min}\). If we add a constant \(c\) to both the maximum and minimum values, \((\text{max}+c)-(\text{min}+c)=\text{max}-\text{min}\). The original range is \(3.2\), so the new range is \(3.2\).

Step5: Effect on IQR

The inter - quartile range \(IQR = Q_3 - Q_1\). When we add a constant \(c\) to each data - point, \(Q_3\) and \(Q_1\) (the third and first quartiles) increase by the same constant \(c\). So, \((Q_3 + c)-(Q_1 + c)=Q_3 - Q_1\). The original \(IQR = 2.4\), so the new \(IQR\) is \(2.4\).

Answer:

Mean: \(13.3\)
Median: \(13.7\)
Mode: \(14\)
Range: \(3.2\)
IQR: \(2.4\)