The coach of the fifth grade girls basketball team measured the height of each player. Their heights in centimeters were $$ 120,122,126,130,133,147,115,106,120 $$, a. Make a stem-and-leaf plot of the players" heights. b. Make a histogram of the players' heights. c. Describe the shape and spread of the data. That is, is it symmetric or non-syimmetric? Does it have more fhan one peak or only one? Is in tighily packed logather or widely spread out? d. Does this data have any outliers? Which measure of center, mean or median, would be appropriate to use to describe the bypical heighte? e. What is the typical height of a player on the team? f. What is the range of the data?

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**a. Stem‐and‐Leaf Plot**

First, list the heights in order:  
$$
106,\;115,\;120,\;120,\;122,\;126,\;130,\;133,\;147
$$

Now, using the tens digit as the stem and the ones digit as the leaf:

$$
\begin{array}{rcl}
10 & | & 6 \quad \text{(for } 106\text{)}\[6mm]
11 & | & 5 \quad \text{(for } 115\text{)}\[6mm]
12 & | & 0 \quad 0 \quad 2 \quad 6 \quad \text{(for } 120,120,122,126\text{)}\[6mm]
13 & | & 0 \quad 3 \quad \text{(for } 130,133\text{)}\[6mm]
14 & | & 7 \quad \text{(for } 147\text{)}
\end{array}
$$

---

**b. Histogram**

We need to decide on bins that cover the range of the data, which extends from $$106$$ to $$147$$. One useful choice is to use 5–unit or 10–unit intervals. For example, choosing 10‐unit bins:

- **Bin 1:** $$105 \leq \text{height} < 115$$  
  Contains: $$106$$  
  Count: $$1$$

- **Bin 2:** $$115 \leq \text{height} < 125$$  
  Contains: $$115,\;120,\;120,\;122$$  
  Count: $$4$$

- **Bin 3:** $$125 \leq \text{height} < 135$$  
  Contains: $$126,\;130,\;133$$  
  Count: $$3$$

- **Bin 4:** $$135 \leq \text{height} < 145$$  
  Contains: None  
  Count: $$0$$

- **Bin 5:** $$145 \leq \text{height} < 155$$  
  Contains: $$147$$  
  Count: $$1$$

A sketch of the histogram would have the height on the horizontal axis marked by these bins and the vertical axis representing frequency (counts). Each bin would have bars with heights corresponding to the counts above.

---

**c. Shape and Spread of the Data**

- **Shape:**  
  The histogram shows a single concentration in Bin 2 ($$115$$–$$125$$) and then fewer observations in the higher bins. This indicates a roughly unimodal distribution with a slight right skew because of the single high value, $$147$$.

- **Spread:**  
  The data range from $$106$$ to $$147$$ (a spread of $$41$$ centimeters). Most of the players’ heights cluster in the lower bins, with a couple of values (one notably high) spread farther away.

---

**d. Outliers and Measure of Center**

- **Outliers:**  
  To check for outliers, one can use the IQR method. The median (middle value) for $$9$$ numbers is the 5th value, which is $$122$$.

Estimate $$Q1$$ (lower quartile) as the median of the lower half $$\{106,115,120,120\}$$:  
  $$
  Q1 = \frac{115+120}{2} = 117.5
  $$
  
  Estimate $$Q3$$ (upper quartile) as the median of the upper half $$\{126,130,133,147\}$$:  
  $$
  Q3 = \frac{130+133}{2} = 131.5
  $$
  
  Then, the interquartile range is:  
  $$
  IQR = Q3 - Q1 = 131.5 - 117.5 = 14
  $$
  
  An observation is flagged as an outlier if it is below  
  $$
  Q1 - 1.5 \cdot IQR = 117.5 - 21 = 96.5
  $$
  or above  
  $$
  Q3 + 1.5 \cdot IQR = 131.5 + 21 = 152.5.
  $$
  
  Since the smallest value, $$106$$, is above $$96.5$$ and the largest value, $$147$$, is below $$152.5$$, there are no formal outliers.

- **Measure of Center:**  
  Even though no strict outliers exist, the slight right skew (due to the high value $$147$$) suggests that the median is a robust measure. Therefore, using the median is appropriate to describe the typical height.

---

**e. Typical Height**

The typical height is best described by the median value. With the data in order:
$$
106,\;115,\;120,\;120,\;122,\;126,\;130,\;133,\;147,
$$
the median (the 5th value) is  
$$
122\text{ cm.}
$$

---

**f. Range of the Data**

The range is the difference between the maximum and minimum values:
$$
\text{Range} = 147 - 106 = 41\text{ cm.}
$$
**a. Stem-and-Leaf Plot** $$ \begin{array}{rcl} 10 & | & 6 \quad \text{(106)}\[6mm] 11 & | & 5 \quad \text{(115)}\[6mm] 12 & | & 0 \quad 0 \quad 2 \quad 6 \quad \text{(120,120,122,126)}\[6mm] 13 & | & 0 \quad 3 \quad \text{(130,133)}\[6mm] 14 & | & 7 \quad \text{(147)} \end{array} $$ **b. Histogram** - **Bin 1:** $$105-115$$ Count: $$1$$ (106) - **Bin 2:** $$115-125$$ Count: $$4$$ (115, 120, 120, 122) - **Bin 3:** $$125-135$$ Count: $$3$$ (126, 130, 133) - **Bin 4:** $$135-145$$ Count: $$0$$ - **Bin 5:** $$145-155$$ Count: $$1$$ (147) **c. Shape and Spread** - **Shape:** Unimodal with a slight right skew due to the high value of $$147$$. - **Spread:** Range from $$106$$ to $$147$$ cm, with most players clustered around $$120$$–$$130$$ cm. **d. Outliers and Measure of Center** - **Outliers:** None detected. - **Measure of Center:** Median ($$122$$ cm) is appropriate due to the slight skew. **e. Typical Height** Median height is $$122$$ cm. **f. Range of the Data** Range is $$41$$ cm ($$147 - 106$$).

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