Question

1. use the stem leaf plot below to find the median and describe the shape of the graph.

baseball throwing distances in feet
stem | leaves
2 | 0 3 5
3 | 1 2 6
4 | 0 2 3 4
5 | 0 3 6 7 8
6 | 1 2 5
7 | 3 4
8 | 2
9 |
10 | 2 5
key: 3 | 2 = 3.2 feet

Explanation:

Step1: Count total data points

First, we count the number of leaves (data points). For stem 2: 3 leaves; stem 3: 3; stem 4: 4; stem 5: 5; stem 6: 3; stem 7: 2; stem 8: 1; stem 9: 1; stem 10: 2. Total = \(3 + 3 + 4 + 5 + 3 + 2 + 1 + 1 + 2 = 24\) data points.

Step2: Find median position

Since \(n = 24\) (even), median is the average of the 12th and 13th values. We list the data points (using key \(3|2 = 3.2\), so stem \(s\), leaf \(l\) is \(s + 0.l\)):
- Stem 2: 2.0, 2.3, 2.5 (3 values, positions 1 - 3)
- Stem 3: 3.1, 3.2, 3.6 (positions 4 - 6)
- Stem 4: 4.0, 4.2, 4.3, 4.4 (positions 7 - 10)
- Stem 5: 5.0, 5.3, 5.6, 5.7, 5.8 (positions 11 - 15) → 12th: 5.3, 13th: 5.6? Wait, no, let's recount positions:
Wait, cumulative counts:
Stem 2: 3 (1 - 3)
Stem 3: 3 (4 - 6) → total 6
Stem 4: 4 (7 - 10) → total 10
Stem 5: 5 (11 - 15) → 11th: 5.0, 12th: 5.3, 13th: 5.6, 14th: 5.7, 15th: 5.8
Wait, no, earlier total was miscalculated. Wait stem 2: 3, stem3:3 (total 6), stem4:4 (10), stem5:5 (15), stem6:3 (18), stem7:2 (20), stem8:1 (21), stem9:1 (22), stem10:2 (24). Yes, \(n = 24\). So median is average of 12th and 13th. Let's list all data points in order:
Stem 2: 2.0, 2.3, 2.5
Stem 3: 3.1, 3.2, 3.6
Stem 4: 4.0, 4.2, 4.3, 4.4
Stem 5: 5.0, 5.3, 5.6, 5.7, 5.8
Stem 6: 6.1, 6.2, 6.5
Stem 7: 7.3, 7.4
Stem 8: 8.2
Stem 9: 9.2
Stem 10: 10.2, 10.5

Now, list positions 1 - 24:
1: 2.0, 2: 2.3, 3: 2.5,
4: 3.1, 5: 3.2, 6: 3.6,
7: 4.0, 8: 4.2, 9: 4.3, 10: 4.4,
11: 5.0, 12: 5.3, 13: 5.6, 14: 5.7, 15: 5.8,
16: 6.1, 17: 6.2, 18: 6.5,
19: 7.3, 20: 7.4,
21: 8.2,
22: 9.2,
23: 10.2, 24: 10.5

So 12th value: 5.3, 13th: 5.6. Median = \(\frac{5.3 + 5.6}{2} = 5.45\) feet.

Step3: Analyze shape (skewness)

To check skewness, we look at the distribution. The left (lower stems) have fewer data points, and the right (higher stems) also have fewer, but let's see the peak. The stem with the most leaves is stem 5 (5 leaves), then stem 4 (4), stem 2/3/6 (3 each). The data has a peak at stem 5, and the tail on the right (higher stems: 7,8,9,10) is a bit longer? Wait, no, actually, the left tail (lower stems: 2,3) has 3 + 3 = 6, middle (4,5,6) has 4 + 5 + 3 = 12, right (7,8,9,10) has 2 + 1 + 1 + 2 = 6. Wait, actually, the distribution is symmetric? Wait no, let's check the number of data points on each side of the median. The median is between 12th and 13th, both in stem 5. Wait, the left of median (positions 1 - 12) and right (13 - 24). Wait, no, for even \(n\), median is average of \(n/2\) and \(n/2 + 1\). Here, \(n = 24\), so \(n/2 = 12\), \(n/2 + 1 = 13\). The 12th is 5.3, 13th is 5.6. Now, looking at the stem - leaf plot, the number of leaves: stem 2 (3), 3 (3), 4 (4), 5 (5), 6 (3), 7 (2), 8 (1), 9 (1), 10 (2). The distribution has a peak at stem 5, and the left (stems 2,3) and right (stems 7,8,9,10) have similar counts, but stem 6 has 3, same as 2,3. Wait, maybe slightly symmetric? Or maybe slightly right - skewed? Wait, no, the right tail (stems 7 - 10) has 2 + 1 + 1 + 2 = 6, left tail (stems 2 - 3) has 3 + 3 = 6, middle (4 - 6) has 4 + 5 + 3 = 12. Wait, actually, the distribution is approximately symmetric, but let's check the mean vs median? Wait, no, the question is about the shape (skewness, symmetric, etc.). Alternatively, maybe the graph is slightly right - skewed because the right tail (higher stems) has some values, but actually, let's count the number of data points below and above the median. The median is 5.45. Data points below 5.45: positions 1 - 12 (values up to 5.3). Data points above 5.45: positions 13 - 24 (values from 5.6 onwards). Wait, no, the 12th is 5.3 (below 5.45), 13th is 5.6 (above). Now, how many data points are below 5.45? Let's list all values:

Below 5.45:
Stem 2: 2.0, 2.3, 2.5 (3)
Stem 3: 3.1, 3.2, 3.6 (3)
Stem 4: 4.0, 4.2, 4.3, 4.4 (4)
Stem 5: 5.0, 5.3 (2) → total: 3 + 3 + 4 + 2 = 12
Above 5.45:
Stem 5: 5.6, 5.7, 5.8 (3)
Stem 6: 6.1, 6.2, 6.5 (3)
Stem 7: 7.3, 7.4 (2)
Stem 8: 8.2 (1)
Stem 9: 9.2 (1)
Stem 10: 10.2, 10.5 (2) → total: 3 + 3 + 2 + 1 + 1 + 2 = 12
So equal number of data points below and above the median, so the graph is symmetric. Wait, but stem 5 has 5 leaves, which is the peak. So the shape is symmetric (or approximately symmetric) with a peak at stem 5 (around 5 feet range).

Answer:

Median: \(5.45\) feet. The shape of the graph is approximately symmetric (or slightly symmetric) with a peak at the stem representing values around 5 feet, and the left and right tails having similar distributions of data points.