Question

daily temperatures for two cities, salem and oxford, were recorded for one week. using statistics, answer the question below! which city had the highest variability in temperatures? s m t w th f s salem 56 61 55 62 58 61 60 oxford 71 52 58 50 89 60 61 salem oxford both had the same variability.

Explanation:

Step1: Recall variance formula

Variance $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}$, where $x_{i}$ are data - points, $\bar{x}$ is the mean and $n$ is the number of data - points. First, find the mean for Salem.
$\bar{x}_{Salem}=\frac{56 + 61+55+62+58+61+60}{7}=\frac{413}{7}=59$

Step2: Calculate squared - differences for Salem

$(56 - 59)^{2}=(-3)^{2}=9$, $(61 - 59)^{2}=2^{2}=4$, $(55 - 59)^{2}=(-4)^{2}=16$, $(62 - 59)^{2}=3^{2}=9$, $(58 - 59)^{2}=(-1)^{2}=1$, $(61 - 59)^{2}=2^{2}=4$, $(60 - 59)^{2}=1^{2}=1$
$\sum_{i = 1}^{7}(x_{i}-\bar{x}_{Salem})^{2}=9 + 4+16+9+1+4+1=44$
$Var_{Salem}=\frac{44}{7}\approx6.29$

Step3: Find the mean for Oxford

$\bar{x}_{Oxford}=\frac{71+52+58+50+89+60+61}{7}=\frac{441}{7}=63$

Step4: Calculate squared - differences for Oxford

$(71 - 63)^{2}=8^{2}=64$, $(52 - 63)^{2}=(-11)^{2}=121$, $(58 - 63)^{2}=(-5)^{2}=25$, $(50 - 63)^{2}=(-13)^{2}=169$, $(89 - 63)^{2}=26^{2}=676$, $(60 - 63)^{2}=(-3)^{2}=9$, $(61 - 63)^{2}=(-2)^{2}=4$
$\sum_{i = 1}^{7}(x_{i}-\bar{x}_{Oxford})^{2}=64 + 121+25+169+676+9+4=1068$
$Var_{Oxford}=\frac{1068}{7}\approx152.57$
Since $Var_{Oxford}>Var_{Salem}$, Oxford has higher variability.

Answer:

Oxford