Directivity refers to the ability of an antenna to focus its signal towards a singular region. Essentially, the more directivity an antenna has, the more directional (and the less isotropic/omni-directional) it becomes. High directivity is great for when you're more likely to know exactly where both your receiver and transmitter will be facing - this configuration provides maximum power and provides a lower bit error rate. However, low directivity is more ideal when it can be difficult to pinpoint the location of your receiver/transmitter, and you require a more broad field of view at the expense of a decreased power signal.

Typical values for directivity range between 1.5 and 2 for dipole antennas, and as large as 100 to 10k for horn and dish antennas. Directivity can also be given in dB, in which case the given value is just 10logx of the original value, x.

Some of the math behind directivity is described below.

Given a radiation pattern, $F(\theta, \phi)$ in spherical coordinates for some particular antenna, directivity can be formally defined as shown below. Note that the radiation pattern $F$ is normalized, such that all the values are scaled so that the maximum power across the radiation pattern is reduced to 1.

$$ D = \frac{1}{\frac{1}{4\pi}\int_{0}^{2\pi}{\int_{0}^{\pi}}{|F(\theta, \phi)|sin\theta d\theta}d\phi} $$

Here, the denominator represents the average power radiated by the antenna across a 3D field.

(Intuitively, this makes sense too: the inner integral performs a partial integration of $Fsin\theta$ with respect to $\theta$ for a given $\phi$ (thereby giving the area of a 2D "circle" within the field of view, and then the outer integral ensures this integration occurs for all $\phi$. If we think of the radiation pattern of an isotropic antenna as being a perfect sphere, $F(\theta, \phi)=1$, we can think of integrating over $\theta$ as taking a 2D circular slice of the 3D sphere and calculating its area, then adding up chunks across the globe as we sweep through the $\phi$ angle. With any other antenna pattern, we're doing essentially the same thing, but with a non-isotropic function for $F$. Finally, we divide by $4\pi$ to get an average).

The numerator on the other hand, represents the maximum value of the radiation pattern function. For an isotropic antenna where $F(\theta, \phi)=1$, the average of the radiated power (denominator) is 1, and so the directivity is 1. Notice that as an antenna becomes more directional, its average power over a 3D field decreases. As a result, the denominator decreases, and the overall directivity increases.

<aside> 💡 Similar to polar coordinates in 2D space, spherical coordinates use two angles and one magnitude (in this case $F$ defined at a particular point) to describe a point on a graph. Usually, $\theta$ represents the angle between the vector (defined by the point and the origin) and the +z axis, and $\phi$ represents the angle between the vector's projection onto the x-y plane and the +x axis. See the adjacent image for further clarification. Note that if we allow one of these angles (namely $\phi$) to take on a domain $[0, 2\pi]$, we only need 180 degrees on the other angle to be able to describe the position of the vector. As such, we limit $\theta \in [0, \pi]$.

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Source: https://www.antenna-theory.com/basics/directivity.php