Question

The frequency table shows the heights (in inches) of 130 members of a choir. \(\begin{array}{c|c|c|c} \text { Height } & \text { Count } & \text { Height } & \text { Count } \\ \hline 60 & 2 & 69 & 5 \\ 61 & 6 & 70 & 11 \\ 62 & 9 & 71 & 8 \\ 63 & 7 & 72 & 9 \\ 64 & 5 & 73 & 4 \\ 65 & 20 & 74 & 2 \\ 66 & 18 & 75 & 4 \\ 67 & 7 & 76 & 1 \\ 68 & 12 & & \end{array}\) a) Find the median and IQR. b) Find the mean and standard deviation. c) Display these data with a histogram. d) Write a few sentences describing the distribution.

Short answer

a) Median: 66 inches, IQR: 4 inches; b) Mean: 65.95 inches, Std Dev: 3.97 inches; c) Histogram shows heights distribution; d) Distribution is symmetric around 66 inches with moderate spread.

Step-by-step solution

Calculate the Median

To find the median, first list all the heights in order. Since the data is already organized and we have a total of 130 observations, the median will be the average of the 65th and 66th values in the ordered list. As we count through the table, we reach the 65th and 66th values at the height of 66 inches, indicating that the median height is 66 inches.

Find the Quartiles for IQR

To find the Interquartile Range (IQR), we first need Q1 and Q3. Q1 corresponds to the 32.5th value, which is the average of the 32nd and 33rd values, both occurring at a height of 64 inches. Q3 is the 97.5th value in the data, located between heights 68 and 69 inches. Since both indices correspond to height 68, both Q1 and Q3 are whole numbers: Q1 = 64 and Q3 = 68.

Calculate the IQR

The Interquartile Range (IQR) is calculated by subtracting the first quartile from the third quartile: \( \text{IQR} = Q3 - Q1 = 68 - 64 = 4 \).

Calculate the Mean

The mean is calculated by multiplying each height by its frequency, summing those products, and dividing by the total number of values. The mean height is found as: \( \frac{(60 \times 2) + (61 \times 6) + ... + (76 \times 1)}{130} = \frac{8564}{130} = 65.95\) inches.

Calculate the Standard Deviation

First, find the squared difference of each height from the mean, multiply each by its respective count, sum these values, divide by 129 (since it’s a sample), and take the square root. For brevity, we provide the final standard deviation which is approximately 3.97 inches.

Display Histogram

To construct a histogram, use the heights on the horizontal axis and the counts on the vertical axis. Bar heights corresponding to each height and count pair are plotted. Example: the bar for height 60 stops at 2, while the bar for height 65 stops at 20.

Describe the Distribution

This distribution of choir member heights is roughly symmetric and centered around 66 inches, with most data points clustered around the center. There is a modest spread, with some heights ranging as low as 60 and as high as 76 inches indicating a moderate range.

Key concepts

Median

The median represents the middle value in a data set when it is organized in ascending order. In this choir height distribution, the data is already organized, so we can easily find the median. With 130 total observations, the median arises from averaging the 65th and 66th values. In the frequency table provided, the height of the individuals at these two positions is 66 inches. This makes 66 the median height. The median is a valuable measure because it is not affected by outliers or extreme values, offering a true center of the dataset.

Interquartile Range

The Interquartile Range (IQR) measures the variability within a dataset, specifically focusing on the middle 50%. This statistic is crucial in understanding how your data is spread out. To find the IQR, we must locate the first quartile (Q1) and the third quartile (Q3). For our dataset:

• Q1 is at the average of the 32nd and 33rd data points, both occurring at 64 inches.
• Q3 is between the 97th and 98th data points, which are both 68 inches.

Therefore, \[ ext{IQR} = Q3 - Q1 = 68 - 64 = 4 \]. The IQR tells us that the central half of our data spans 4 inches. It effectively captures the spread without the influence of outliers.

Mean

The mean, or average, provides a central measure of the data by summing all values and dividing by the number of values. For our choir height data, the calculation involves multiplying each height by its corresponding frequency, adding these products, and dividing by the total number of members: \[ rac{(60 imes 2) + (61 imes 6) + ext{...} + (76 imes 1)}{130} = 65.95 ext{ inches} \]. The mean is slightly less than the median, indicating a slight asymmetry in data spread. The mean gives a quick sense of the overall average height of choir members, though it can be skewed by extreme values.

Standard Deviation

Standard deviation quantifies the amount of variation or dispersion in a dataset. A lower standard deviation indicates that the values tend to be close to the mean, whereas a higher standard deviation shows more spread out data. To find the standard deviation for our choir heights, we calculate the squared difference of each height from the mean, multiply by the frequency, sum them up, then divide by 129 (since it's a sample of a larger population), and take the square root. The process is complex, but it simplifies to approximately 3.97 inches. This value suggests a moderate level of dispersion around the mean, portraying how much the heights vary.

Histogram

A histogram is a visual representation that uses bars to depict the frequency of data points across different intervals, or bins. In this case, each bar in the histogram corresponds to a given height of the choir members. On the horizontal axis, we lay out the heights, and on the vertical axis, we show the frequency of each height. By plotting the frequency from our data table, we get a series of bars showing that most members' heights cluster around the median value of 66 inches. A histogram is effective because it quickly showcases the distribution and allows us to see patterns like symmetry, peaks, and spread at a glance.