According to the 2003 Annual Consumer Spending Survey, the...

According to the 2003 Annual Consumer Spending Survey, the average monthly Bank of America Visa credit card charge was $\$ 1838$ (U.S. Airways Attaché Magazine, December 2003 ). A sample of monthly credit card charges provides the following data. \[ \begin{array}{rrrrr} 236 & 1710 & 1351 & 825 & 7450 \\ 316 & 4135 & 1333 & 1584 & 387 \\ 991 & 3396 & 170 & 1428 & 1688 \end{array} \] a. Compute the mean and median. b. Compute the first and third quartiles. c. $\quad$ Compute the range and interquartile range. d. Compute the variance and standard deviation. e. The skewness measure for these data is $2.12 .$ Comment on the shape of this distribution. Is it the shape you would expect? Why or why not? f. Do the data contain outliers?

According to the 2003 Annual Consumer Spending Survey, the average monthly Bank of America Visa credit card charge was $\$ 1838$ (U.S. Airways Attaché Magazine, December 2003 ). A sample of monthly credit card charges provides the following data.
\[
\begin{array}{rrrrr}
236 & 1710 & 1351 & 825 & 7450 \\
316 & 4135 & 1333 & 1584 & 387 \\
991 & 3396 & 170 & 1428 & 1688
\end{array}
\]
a. Compute the mean and median.
b. Compute the first and third quartiles.
c. $\quad$ Compute the range and interquartile range.
d. Compute the variance and standard deviation.
e. The skewness measure for these data is $2.12 .$ Comment on the shape of this distribution. Is it the shape you would expect? Why or why not?
f. Do the data contain outliers?

Key Concepts

Skewness

Skewness is a measure of the asymmetry of the data distribution. A positive skew implies a longer or fatter tail on the right side, while a negative skew implies the opposite. Understanding skewness helps in assessing the distribution shape and potential deviations from normality.

Outliers

Outliers are observations that lie far from the other data points, often defined using a threshold such as 1.5 times the interquartile range below Q1 or above Q3. They can signal data entry errors, variability in measurement, or genuine variability in the data set.

Standard Deviation

Standard deviation is the square root of the variance and measures the average distance of data points from the mean. It is expressed in the same units as the data, making it more intuitive for understanding variability.

Variance

Variance quantifies the average squared deviations of each data point from the mean. It provides an overall measure of data spread, with higher values indicating more dispersion in the data set.

Median

The median is the middle value in a sorted data set, which divides the data into two equal halves. It is a robust measure of central tendency, especially useful when the data distribution is skewed or contains outliers.

Range

The range is the difference between the maximum and minimum values in a data set, providing a simple measure of the overall spread. It is sensitive to extreme values and may not represent the typical dispersion of the data.

Quartiles

Quartiles are values that divide the sorted data set into four equal parts. The first quartile (Q1) is the median of the lower half of the data, and the third quartile (Q3) is the median of the upper half. They help in understanding the spread and central concentration of the middle portion of the data.

Mean

The mean is a measure of central tendency calculated by summing all the values in a data set and then dividing by the number of observations. It provides a representative value for the data, but can be affected by extreme values (outliers).

Interquartile Range (IQR)

The interquartile range is the difference between the third quartile (Q3) and first quartile (Q1). It measures the spread of the middle 50% of the data and is less affected by outliers, offering a better sense of the 'typical' dispersion in the data.

Transcript

00:01 Hello, so here, part a, we have our given data is 236, 3116, 9991, and so on.

00:07 And the last value is 1 ,688.

00:10 You have a number of, we have n equals 15 here.

00:13 So we can put our data in ascending order.

00:14 We go from 170 to 236, 316, 387, and so on, up to our largest value of 7 ,450.

00:22 And then, our mean is the sum of each xi divided by n.

00:26 So that's 27 ,000 then divided by 15, which is going to be equal.

00:30 To 1800.

00:33 And we know the 50th percentile then is the median.

00:36 So we have i is going to be equal to p over 100 times n.

00:41 That's equal to 50 over 100 times 15, which is equal to 7 .5.

00:48 So here i is not an integer.

00:50 So we round up.

00:51 And then the position of the 50th percentile is the next integer greater than 7 .5.

00:57 So that's the eighth position.

00:58 And then from our data, we have the median is the value in the eighth position, which is going to be 1 ,351.

01:10 And then for part b, we know that the 25th percentile is the first quartile.

01:16 So that's going to be 3 .75, which we have then round up to the fourth position.

01:21 And then we have the data value in the fourth position is 387.

01:27 And the 75th percentile is the third quartile.

01:30 So we get that, that's 11 .25.

01:36 Again, we're not in integers.

01:37 We round up.

01:38 And the position of the 75th percentile is the next integer greater than 11 .25.

01:44 So that's going to be in the 12th position.

01:47 And from our data we have at the 75th percentile is the data in the 12th position, which is going to be 1 ,710...

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